id stringlengths 11 15 | title stringlengths 7 171 | problem stringlengths 9 4.33k | solution stringlengths 6 19k | problem_wikitext stringlengths 9 4.42k | solution_wikitext stringlengths 7 19.1k | proof_title stringlengths 9 171 | theorem_url stringlengths 34 198 | proof_url stringlengths 36 198 | categories listlengths 0 9 | theorem_references listlengths 0 36 | proof_references listlengths 0 253 |
|---|---|---|---|---|---|---|---|---|---|---|---|
proofwiki-3400 | Urysohn Space is T2.5 | Let $\struct {S, \tau}$ be an Urysohn space.
Then $\struct {S, \tau}$ is also a $T_{2 \frac 1 2}$ space. | Let $T = \struct {S, \tau}$ be an Urysohn space.
{{Recall|Urysohn Space|Urysohn space}}
{{:Definition:Urysohn Space}}
{{Recall|T2.5 Space|$T_{2 \frac 1 2}$ space}}
{{:Definition:T2.5 Space/Definition 1}}
{{proof wanted|Then we do some stuff.}}
Thus:
:$\forall x, y \in S: x \ne y: \exists U, V \in \tau: x \in U, y \in V... | Let $\struct {S, \tau}$ be an [[Definition:Urysohn Space|Urysohn space]].
Then $\struct {S, \tau}$ is also a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]]. | Let $T = \struct {S, \tau}$ be an [[Definition:Urysohn Space|Urysohn space]].
{{Recall|Urysohn Space|Urysohn space}}
{{:Definition:Urysohn Space}}
{{Recall|T2.5 Space|$T_{2 \frac 1 2}$ space}}
{{:Definition:T2.5 Space/Definition 1}}
{{proof wanted|Then we do some stuff.}}
Thus:
:$\forall x, y \in S: x \ne y: \exis... | Urysohn Space is T2.5 | https://proofwiki.org/wiki/Urysohn_Space_is_T2.5 | https://proofwiki.org/wiki/Urysohn_Space_is_T2.5 | [
"T2.5 Spaces",
"Urysohn Spaces"
] | [
"Definition:Urysohn Space",
"Definition:T2.5 Space"
] | [
"Definition:Urysohn Space",
"Definition:T2.5 Space"
] |
proofwiki-3401 | Perfectly Normal Space is Completely Normal | Let $T = \struct {S, \tau}$ be a perfectly normal space.
Then $T$ is also a completely normal space. | Let $T = \struct {S, \tau}$ be a perfectly normal space.
{{Recall|Perfectly Normal Space|perfectly normal space|index = 1}}
{{:Definition:Perfectly Normal Space/Definition 1}}
{{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
From Perfectly $T_4$ S... | Let $T = \struct {S, \tau}$ be a [[Definition:Perfectly Normal Space|perfectly normal space]].
Then $T$ is also a [[Definition:Completely Normal Space|completely normal space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Perfectly Normal Space|perfectly normal space]].
{{Recall|Perfectly Normal Space|perfectly normal space|index = 1}}
{{:Definition:Perfectly Normal Space/Definition 1}}
{{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Sp... | Perfectly Normal Space is Completely Normal | https://proofwiki.org/wiki/Perfectly_Normal_Space_is_Completely_Normal | https://proofwiki.org/wiki/Perfectly_Normal_Space_is_Completely_Normal | [
"Completely Normal Spaces",
"Perfectly Normal Spaces"
] | [
"Definition:Perfectly Normal Space",
"Definition:Completely Normal Space"
] | [
"Definition:Perfectly Normal Space",
"Perfectly T4 Space is T5",
"Definition:T5 Space"
] |
proofwiki-3402 | Open Set in T3 Space is Union of Regular Open Sets | Let $T = \struct {S, \tau}$ be a $T_3$ space.
Then every open set of $T$ can be expressed as the union of regular open sets. | Let $\BB = \set {B \subseteq S: B^{- \circ} = B}$.
In other words, $\BB$ is the collection of regular open sets contained in $S$.
It follows immediately from the definition of a basis that our theorem is proved if we can show that:
:$\BB$ is a cover for $S$
:$\forall U, V \in \BB: \forall x \in U \cap V: \exists W \in ... | Let $T = \struct {S, \tau}$ be a [[Definition:T3 Space|$T_3$ space]].
Then every [[Definition:Open Set (Topology)|open set]] of $T$ can be expressed as the [[Definition:Set Union|union]] of [[Definition:Regular Open Set|regular open sets]]. | Let $\BB = \set {B \subseteq S: B^{- \circ} = B}$.
In other words, $\BB$ is the collection of [[Definition:Regular Open Set|regular open sets]] contained in $S$.
It follows immediately from the definition of a [[Definition:Synthetic Basis|basis]] that our theorem is proved if we can show that:
:$\BB$ is a [[Definitio... | Open Set in T3 Space is Union of Regular Open Sets | https://proofwiki.org/wiki/Open_Set_in_T3_Space_is_Union_of_Regular_Open_Sets | https://proofwiki.org/wiki/Open_Set_in_T3_Space_is_Union_of_Regular_Open_Sets | [
"T3 Spaces",
"Regular Open Sets"
] | [
"Definition:T3 Space",
"Definition:Open Set/Topology",
"Definition:Set Union",
"Definition:Regular Open Set"
] | [
"Definition:Regular Open Set",
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Cover of Set",
"Definition:Cover of Set",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Set is Subset of Union/General Result",
"Definition:Cover of Set",
"Definition:Synthetic Basis/Defini... |
proofwiki-3403 | Regular Space is Semiregular | Let $\struct {S, \tau}$ be a regular space.
Then $\struct {S, \tau}$ is also a semiregular space. | Let $T = \struct {S, \tau}$ be a regular space.
{{Recall|Regular Space|index = 3}}
{{:Definition:Regular Space/Definition 3}}
{{Recall|Semiregular Space|semiregular space}}
{{:Definition:Semiregular Space}}
We also have that an open set in a $T_3$ space is the union of regular open sets.
That is, from the definition of... | Let $\struct {S, \tau}$ be a [[Definition:Regular Space|regular space]].
Then $\struct {S, \tau}$ is also a [[Definition:Semiregular Space|semiregular space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Regular Space|regular space]].
{{Recall|Regular Space|index = 3}}
{{:Definition:Regular Space/Definition 3}}
{{Recall|Semiregular Space|semiregular space}}
{{:Definition:Semiregular Space}}
We also have that an [[Open Set in T3 Space is Union of Regular Open Sets|open se... | Regular Space is Semiregular | https://proofwiki.org/wiki/Regular_Space_is_Semiregular | https://proofwiki.org/wiki/Regular_Space_is_Semiregular | [
"Semiregular Spaces",
"Regular Spaces"
] | [
"Definition:Regular Space",
"Definition:Semiregular Space"
] | [
"Definition:Regular Space",
"Open Set in T3 Space is Union of Regular Open Sets",
"Definition:Basis (Topology)",
"Definition:Regular Open Set",
"Definition:Basis (Topology)"
] |
proofwiki-3404 | Compact Space is Sigma-Compact | Every '''compact space''' is also a '''$\sigma$-compact space'''. | By definition, a $\sigma$-compact space is the union of countably many compact sets.
A compact space is the union of exactly one compact space.
Hence the result.
{{qed}} | Every '''[[Definition:Compact Topological Space|compact space]]''' is also a '''[[Definition:Sigma-Compact Space|$\sigma$-compact space]]'''. | By definition, a [[Definition:Sigma-Compact Space|$\sigma$-compact space]] is the [[Definition:Set Union|union]] of [[Definition:Countable Set|countably]] many [[Definition:Compact Topological Space|compact]] sets.
A [[Definition:Compact Topological Space|compact space]] is the [[Definition:Set Union|union]] of exactl... | Compact Space is Sigma-Compact | https://proofwiki.org/wiki/Compact_Space_is_Sigma-Compact | https://proofwiki.org/wiki/Compact_Space_is_Sigma-Compact | [
"Compact Topological Spaces",
"Sigma-Compact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Compact Topological Space",
"Definition:Sigma-Compact Space"
] | [
"Definition:Sigma-Compact Space",
"Definition:Set Union",
"Definition:Countable Set",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space",
"Definition:Set Union",
"Definition:Compact Topological Space"
] |
proofwiki-3405 | Sigma-Compact Space is Lindelöf | Every $\sigma$-compact space is a Lindelöf space. | Let $T = \struct {S, \tau}$ be a $\sigma$-compact space.
{{Recall|Lindelöf Space|Lindelöf space}}
{{:Definition:Lindelöf Space}}
{{Recall|Sigma-Compact Space|$\sigma$-compact space}}
{{:Definition:Sigma-Compact Space}}
So by definition of $\sigma$-compact space:
:$T = \bigcup \HH$
where $\HH$ is a countable set of comp... | Every [[Definition:Sigma-Compact Space|$\sigma$-compact space]] is a [[Definition:Lindelöf Space|Lindelöf space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Sigma-Compact Space|$\sigma$-compact space]].
{{Recall|Lindelöf Space|Lindelöf space}}
{{:Definition:Lindelöf Space}}
{{Recall|Sigma-Compact Space|$\sigma$-compact space}}
{{:Definition:Sigma-Compact Space}}
So by definition of [[Definition:Sigma-Compact Space|$\sigma$-c... | Sigma-Compact Space is Lindelöf | https://proofwiki.org/wiki/Sigma-Compact_Space_is_Lindelöf | https://proofwiki.org/wiki/Sigma-Compact_Space_is_Lindelöf | [
"Sigma-Compact Spaces",
"Lindelöf Spaces",
"Sequence of Implications of Global Compactness Properties",
"Sequence of Implications of Metric Space Compactness Properties"
] | [
"Definition:Sigma-Compact Space",
"Definition:Lindelöf Space"
] | [
"Definition:Sigma-Compact Space",
"Definition:Sigma-Compact Space",
"Definition:Countable Set",
"Definition:Compact Topological Space/Subspace",
"Definition:Open Cover",
"Definition:Compact Topological Space",
"Definition:Cover of Set",
"Definition:Finite Set",
"Definition:Element",
"Definition:Co... |
proofwiki-3406 | Sequentially Compact Space is Countably Compact | A sequentially compact topological space is also countably compact. | {{Recall|Countably Compact Space|countably compact space|index = 3}}
{{:Definition:Countably Compact Space/Definition 3}}
Let $T = \struct {S, \tau}$ be a sequentially compact topological space.
{{Recall|Sequentially Compact Space|sequentially compact space}}
{{:Definition:Sequentially Compact Space}}
The result follow... | A [[Definition:Sequentially Compact Space|sequentially compact]] [[Definition:Topological Space|topological space]] is also [[Definition:Countably Compact Space|countably compact]]. | {{Recall|Countably Compact Space|countably compact space|index = 3}}
{{:Definition:Countably Compact Space/Definition 3}}
Let $T = \struct {S, \tau}$ be a [[Definition:Sequentially Compact Space|sequentially compact]] [[Definition:Topological Space|topological space]].
{{Recall|Sequentially Compact Space|sequentially... | Sequentially Compact Space is Countably Compact/Proof 1 | https://proofwiki.org/wiki/Sequentially_Compact_Space_is_Countably_Compact | https://proofwiki.org/wiki/Sequentially_Compact_Space_is_Countably_Compact/Proof_1 | [
"Sequentially Compact Space is Countably Compact",
"Sequentially Compact Spaces",
"Countably Compact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Sequentially Compact Space",
"Definition:Topological Space",
"Definition:Countably Compact Space"
] | [
"Definition:Sequentially Compact Space",
"Definition:Topological Space",
"Limit of Sequence is Accumulation Point"
] |
proofwiki-3407 | Sequentially Compact Space is Countably Compact | A sequentially compact topological space is also countably compact. | {{Recall|Countably Compact Space|countably compact space|index = 1}}
{{:Definition:Countably Compact Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a sequentially compact topological space.
{{Recall|Sequentially Compact Space|sequentially compact space}}
{{:Definition:Sequentially Compact Space}}
{{AimForCont}} $T... | A [[Definition:Sequentially Compact Space|sequentially compact]] [[Definition:Topological Space|topological space]] is also [[Definition:Countably Compact Space|countably compact]]. | {{Recall|Countably Compact Space|countably compact space|index = 1}}
{{:Definition:Countably Compact Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a [[Definition:Sequentially Compact Space|sequentially compact]] [[Definition:Topological Space|topological space]].
{{Recall|Sequentially Compact Space|sequentially... | Sequentially Compact Space is Countably Compact/Proof 2 | https://proofwiki.org/wiki/Sequentially_Compact_Space_is_Countably_Compact | https://proofwiki.org/wiki/Sequentially_Compact_Space_is_Countably_Compact/Proof_2 | [
"Sequentially Compact Space is Countably Compact",
"Sequentially Compact Spaces",
"Countably Compact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Sequentially Compact Space",
"Definition:Topological Space",
"Definition:Countably Compact Space"
] | [
"Definition:Sequentially Compact Space",
"Definition:Topological Space",
"Definition:Countably Compact Space",
"Definition:Cover of Set/Countable",
"Definition:Open Cover",
"Definition:Subcover/Finite",
"Definition:Arbitrary",
"Definition:Sequence/Infinite Sequence",
"Definition:Subsequence",
"Def... |
proofwiki-3408 | Countably Compact Space is Weakly Countably Compact | Every countably compact space is a weakly countably compact space. | Let $T = \struct {S, \tau}$ be a countably compact space.
{{Recall|Weakly Countably Compact Space|weakly countably compact}}
{{:Definition:Weakly Countably Compact Space}}
{{Recall|Countably Compact Space|countably compact space|index = 1}}
{{:Definition:Countably Compact Space/Definition 1}}
Let $A \subseteq S$ be cou... | Every [[Definition:Countably Compact Space|countably compact space]] is a [[Definition:Weakly Countably Compact Space|weakly countably compact space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Countably Compact Space|countably compact space]].
{{Recall|Weakly Countably Compact Space|weakly countably compact}}
{{:Definition:Weakly Countably Compact Space}}
{{Recall|Countably Compact Space|countably compact space|index = 1}}
{{:Definition:Countably Compact Space... | Countably Compact Space is Weakly Countably Compact/Proof 2 | https://proofwiki.org/wiki/Countably_Compact_Space_is_Weakly_Countably_Compact | https://proofwiki.org/wiki/Countably_Compact_Space_is_Weakly_Countably_Compact/Proof_2 | [
"Countably Compact Space is Weakly Countably Compact",
"Countably Compact Spaces",
"Weakly Countably Compact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Countably Compact Space",
"Definition:Weakly Countably Compact Space"
] | [
"Definition:Countably Compact Space",
"Definition:Countably Infinite",
"Definition:Limit Point/Topology/Set",
"Set with no Limit Point is Closed",
"Definition:Closed Set/Topology",
"Open Neighborhood of Element of Set with no Limit Points",
"Definition:Open Set/Topology",
"Definition:Set",
"Definiti... |
proofwiki-3409 | T1 Space is Weakly Countably Compact iff Countably Compact | Let $T = \struct {S, \tau}$ be a $T_1$ space.
Then $T$ is weakly countably compact {{iff}} $T$ is countably compact. | === Sufficient Condition ===
We have that a Countably Compact Space is Weakly Countably Compact whatever the separation properties.
{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:T1 Space|$T_1$ space]].
Then $T$ is [[Definition:Weakly Countably Compact Space|weakly countably compact]] {{iff}} $T$ is [[Definition:Countably Compact Space|countably compact]]. | === Sufficient Condition ===
We have that a [[Countably Compact Space is Weakly Countably Compact]] whatever the [[Definition:Separation Axioms|separation properties]].
{{qed|lemma}} | T1 Space is Weakly Countably Compact iff Countably Compact | https://proofwiki.org/wiki/T1_Space_is_Weakly_Countably_Compact_iff_Countably_Compact | https://proofwiki.org/wiki/T1_Space_is_Weakly_Countably_Compact_iff_Countably_Compact | [
"T1 Spaces",
"Countably Compact Spaces",
"Weakly Countably Compact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:T1 Space",
"Definition:Weakly Countably Compact Space",
"Definition:Countably Compact Space"
] | [
"Countably Compact Space is Weakly Countably Compact",
"Definition:Tychonoff Separation Axioms"
] |
proofwiki-3410 | Countably Compact Space is Pseudocompact | Let $T = \struct {S, \tau}$ be a countably compact space.
Then $T$ is a pseudocompact space. | {{Recall|Pseudocompact Space|pseudocompact space}}
{{:Definition:Pseudocompact Space}}
Let $T = \struct {S, \tau}$ be a countably compact space.
{{Recall|Countably Compact Space|countably compact space}}
{{:Definition:Countably Compact Space/Definition 1}}
Let $f: S \to \R$ be an arbitrary continuous real-valued functi... | Let $T = \struct {S, \tau}$ be a [[Definition:Countably Compact Space|countably compact space]].
Then $T$ is a [[Definition:Pseudocompact Space|pseudocompact space]]. | {{Recall|Pseudocompact Space|pseudocompact space}}
{{:Definition:Pseudocompact Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Countably Compact Space|countably compact space]].
{{Recall|Countably Compact Space|countably compact space}}
{{:Definition:Countably Compact Space/Definition 1}}
Let $f: S \to \R$ be ... | Countably Compact Space is Pseudocompact | https://proofwiki.org/wiki/Countably_Compact_Space_is_Pseudocompact | https://proofwiki.org/wiki/Countably_Compact_Space_is_Pseudocompact | [
"Countably Compact Spaces",
"Pseudocompact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Countably Compact Space",
"Definition:Pseudocompact Space"
] | [
"Definition:Countably Compact Space",
"Definition:Arbitrary",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Real-Valued Function",
"Definition:Set",
"Definition:Indexing Set/Family of Sets",
"Definition:Subset",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Axiom of Ar... |
proofwiki-3411 | Normal Space is Pseudocompact iff Countably Compact | Let $T = \struct {S, \tau}$ be a normal space.
Then $T$ is pseudocompact {{iff}} $T$ is countably compact. | Let $T = \struct {S, \tau}$ be a normal space.
By Countably Compact Space is Pseudocompact we already have that if $T$ is countably compact then $T$ is pseudocompact.
It remains to prove that if $T$ is pseudocompact then $T$ is countably compact.
{{AimForCont}} $T$ is not countably compact.
By definition of countably c... | Let $T = \struct {S, \tau}$ be a [[Definition:Normal Space|normal space]].
Then $T$ is [[Definition:Pseudocompact Space|pseudocompact]] {{iff}} $T$ is [[Definition:Countably Compact Space|countably compact]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Normal Space|normal space]].
By [[Countably Compact Space is Pseudocompact]] we already have that if $T$ is [[Definition:Countably Compact Space|countably compact]] then $T$ is [[Definition:Pseudocompact Space|pseudocompact]].
It remains to prove that if $T$ is [[Definiti... | Normal Space is Pseudocompact iff Countably Compact | https://proofwiki.org/wiki/Normal_Space_is_Pseudocompact_iff_Countably_Compact | https://proofwiki.org/wiki/Normal_Space_is_Pseudocompact_iff_Countably_Compact | [
"Countably Compact Spaces",
"Pseudocompact Spaces",
"Normal Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Normal Space",
"Definition:Pseudocompact Space",
"Definition:Countably Compact Space"
] | [
"Definition:Normal Space",
"Countably Compact Space is Pseudocompact",
"Definition:Countably Compact Space",
"Definition:Pseudocompact Space",
"Definition:Pseudocompact Space",
"Definition:Countably Compact Space",
"Definition:Countably Compact Space",
"Definition:Countably Compact Space/Definition 5"... |
proofwiki-3412 | Countably Compact Lindelöf Space is Compact | Let $T = \struct {S, \tau}$ be a topological space.
Then:
:$T$ is a compact space
{{iff}}:
:$T$ is both:
::a Lindelöf space
::a countably compact space. | === Sufficient Condition ===
Let $T$ be a compact space.
Then from:
:Compact Space is Countably Compact
:Compact Space is Lindelöf
it follows that $T$ is both a Lindelöf space and a countably compact space.
{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then:
:$T$ is a [[Definition:Compact Topological Space|compact space]]
{{iff}}:
:$T$ is both:
::a [[Definition:Lindelöf Space|Lindelöf space]]
::a [[Definition:Countably Compact Space|countably compact space]]. | === Sufficient Condition ===
Let $T$ be a [[Definition:Compact Topological Space|compact space]].
Then from:
:[[Compact Space is Countably Compact]]
:[[Compact Space is Lindelöf]]
it follows that $T$ is both a [[Definition:Lindelöf Space|Lindelöf space]] and a [[Definition:Countably Compact Space|countably compact s... | Countably Compact Lindelöf Space is Compact | https://proofwiki.org/wiki/Countably_Compact_Lindelöf_Space_is_Compact | https://proofwiki.org/wiki/Countably_Compact_Lindelöf_Space_is_Compact | [
"Countably Compact Spaces",
"Lindelöf Spaces",
"Compact Topological Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Topological Space",
"Definition:Compact Topological Space",
"Definition:Lindelöf Space",
"Definition:Countably Compact Space"
] | [
"Definition:Compact Topological Space",
"Compact Space is Countably Compact",
"Compact Space is Lindelöf",
"Definition:Lindelöf Space",
"Definition:Countably Compact Space",
"Definition:Lindelöf Space",
"Definition:Countably Compact Space",
"Definition:Compact Topological Space"
] |
proofwiki-3413 | Big Picard Theorem | Let:
:$U \subset \C$ be a domain
:$z_0 \in U$
:$f: U \setminus \set {z_0} \to \C$ be a holomorphic function with an essential singularity at $z_0$.
Then $f$ omits at most one complex value $a \in \C$. | {{MissingLinks}}
{{tidy|get rid of the digressional material}}
Montel's Theorem is a very close relative of Picard's theorem.
It states that a family of holomorphic functions that all omit the same two values is normal.
(Recall that a family $\FF$ of holomorphic functions $f: U \to \C$ is normal if every sequence in $\... | Let:
:$U \subset \C$ be a [[Definition:Domain of Mapping|domain]]
:$z_0 \in U$
:$f: U \setminus \set {z_0} \to \C$ be a [[Definition:Holomorphic Function|holomorphic function]] with an [[Definition:Isolated Singularity#Essential Singularity|essential singularity]] at $z_0$.
Then $f$ omits at most one complex value $a ... | {{MissingLinks}}
{{tidy|get rid of the digressional material}}
[[Montel's Theorem]] is a very close relative of Picard's theorem.
It states that a family of holomorphic functions that all omit the same two values is normal.
(Recall that a family $\FF$ of holomorphic functions $f: U \to \C$ is normal if every sequenc... | Big Picard Theorem | https://proofwiki.org/wiki/Big_Picard_Theorem | https://proofwiki.org/wiki/Big_Picard_Theorem | [
"Complex Analysis",
"Essential Singularities"
] | [
"Definition:Domain (Set Theory)/Mapping",
"Definition:Holomorphic Function",
"Definition:Isolated Singularity"
] | [
"Montel's Theorem",
"Definition:Analytic Function",
"Definition:Isolated Singularity/Pole",
"Definition:Isolated Singularity/Pole",
"Definition:Convergent Mapping",
"Montel's Theorem",
"Fundamental Normality Test",
"Definition:Locally Uniform Convergence",
"Maximum Modulus Principle",
"Definition:... |
proofwiki-3414 | Compact Space is Weakly Locally Compact | Let $T = \struct {S, \tau}$ be a compact topological space.
Then $T$ is a weakly locally compact. | Let $T$ be a compact space.
By definition, $T$ is weakly locally compact {{iff}} each point of $S$ has a compact neighborhood in $T$.
Let $x \in S$ be an arbitrary point of $S$.
By Topological Space is Neighborhood of all its Points, $S$ is a neighborhood of $x$.
Thus $x$ has a compact neighborhood in $T$.
Hence the re... | Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact topological space]].
Then $T$ is a [[Definition:Weakly Locally Compact Space|weakly locally compact]]. | Let $T$ be a [[Definition:Compact Topological Space|compact space]].
By definition, $T$ is [[Definition:Weakly Locally Compact Space|weakly locally compact]] {{iff}} each point of $S$ has a [[Definition:Compact Topological Space|compact]] [[Definition:Neighborhood of Point|neighborhood]] in $T$.
Let $x \in S$ be an ... | Compact Space is Weakly Locally Compact/Proof 1 | https://proofwiki.org/wiki/Compact_Space_is_Weakly_Locally_Compact | https://proofwiki.org/wiki/Compact_Space_is_Weakly_Locally_Compact/Proof_1 | [
"Compact Space is Weakly Locally Compact",
"Weakly Locally Compact Spaces",
"Compact Topological Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Compact Topological Space",
"Definition:Weakly Locally Compact Space"
] | [
"Definition:Compact Topological Space",
"Definition:Weakly Locally Compact Space",
"Definition:Compact Topological Space",
"Definition:Neighborhood (Topology)/Point",
"Topological Space is Neighborhood of all its Points",
"Definition:Neighborhood (Topology)/Point",
"Definition:Compact Topological Space"... |
proofwiki-3415 | Compact Space is Weakly Locally Compact | Let $T = \struct {S, \tau}$ be a compact topological space.
Then $T$ is a weakly locally compact. | Follows directly from:
:Compact Space is Weakly $\sigma$-Locally Compact
:a weakly $\sigma$-locally compact space is weakly locally compact by definition.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact topological space]].
Then $T$ is a [[Definition:Weakly Locally Compact Space|weakly locally compact]]. | Follows directly from:
:[[Compact Space is Weakly Sigma-Locally Compact|Compact Space is Weakly $\sigma$-Locally Compact]]
:a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space]] is [[Definition:Weakly Locally Compact Space|weakly locally compact]] by definition.
{{qed}} | Compact Space is Weakly Locally Compact/Proof 3 | https://proofwiki.org/wiki/Compact_Space_is_Weakly_Locally_Compact | https://proofwiki.org/wiki/Compact_Space_is_Weakly_Locally_Compact/Proof_3 | [
"Compact Space is Weakly Locally Compact",
"Weakly Locally Compact Spaces",
"Compact Topological Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Compact Topological Space",
"Definition:Weakly Locally Compact Space"
] | [
"Compact Space is Weakly Sigma-Locally Compact",
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:Weakly Locally Compact Space"
] |
proofwiki-3416 | Completed Riemann Zeta Function has Order One | The completed Riemann zeta function $\xi$ has order at most $1$. | We are required to prove that:
:$\map \xi s = \dfrac 1 2 s \paren {s - 1} \pi^{-s/2} \map \Gamma {\dfrac s 2} \map \zeta s \ll \map \exp {\size s^\beta}$
for all $\beta > 1$, where $\ll$ is the order notation.
Note that by the Functional Equation for Riemann Zeta Function, it is sufficient to check this for $\map \Re s... | The [[Definition:Completed Riemann Zeta Function|completed Riemann zeta function]] $\xi$ has [[Definition:Order of Entire Function|order]] at most $1$. | We are required to prove that:
:$\map \xi s = \dfrac 1 2 s \paren {s - 1} \pi^{-s/2} \map \Gamma {\dfrac s 2} \map \zeta s \ll \map \exp {\size s^\beta}$
for all $\beta > 1$, where $\ll$ is the [[Definition:Big-O Notation|order notation]].
Note that by the [[Functional Equation for Riemann Zeta Function]], it is suf... | Completed Riemann Zeta Function has Order One | https://proofwiki.org/wiki/Completed_Riemann_Zeta_Function_has_Order_One | https://proofwiki.org/wiki/Completed_Riemann_Zeta_Function_has_Order_One | [
"Riemann Zeta Function"
] | [
"Definition:Completed Riemann Zeta Function",
"Definition:Order of Entire Function"
] | [
"Definition:Big-O Notation",
"Functional Equation for Riemann Zeta Function",
"Definition:Gamma Function",
"Stirling's Formula for Gamma Function",
"Definition:Pole",
"Integral Representation of Riemann Zeta Function in terms of Fractional Part",
"Upper Bound of Natural Logarithm",
"Category:Riemann Z... |
proofwiki-3417 | Product Equation for Riemann Zeta Function | There exists a constant $B$ such that:
:$\ds \frac {\map {\zeta'} s} {\map \zeta s} = B - \frac 1 {s - 1} + \frac 1 2 \ln \pi - \frac 1 2 \frac {\map {\Gamma'} {s / 2 + 1} } {\map \Gamma {s / 2 + 1} } + \sum_\rho \paren {\frac 1 {s - \rho} + \frac 1 \rho}$
where:
:$\zeta$ is the Riemann zeta function
:$\rho$ runs over ... | Let $\xi$ be the completed Riemann zeta function:
{{begin-eqn}}
{{eqn | l = \map \xi s
| r = \frac 1 2 s \paren {s - 1} \pi^{-s / 2} \map \Gamma {\frac s 2} \map \zeta s
| c = {{Defof|Completed Riemann Zeta Function}}
}}
{{eqn | r = \paren {s - 1} \pi^{-s / 2} \map \Gamma {\frac s 2 + 1} \map \zeta s
... | There exists a constant $B$ such that:
:$\ds \frac {\map {\zeta'} s} {\map \zeta s} = B - \frac 1 {s - 1} + \frac 1 2 \ln \pi - \frac 1 2 \frac {\map {\Gamma'} {s / 2 + 1} } {\map \Gamma {s / 2 + 1} } + \sum_\rho \paren {\frac 1 {s - \rho} + \frac 1 \rho}$
where:
:$\zeta$ is the [[Definition:Riemann Zeta Function|Rie... | Let $\xi$ be the [[Definition:Completed Riemann Zeta Function|completed Riemann zeta function]]:
{{begin-eqn}}
{{eqn | l = \map \xi s
| r = \frac 1 2 s \paren {s - 1} \pi^{-s / 2} \map \Gamma {\frac s 2} \map \zeta s
| c = {{Defof|Completed Riemann Zeta Function}}
}}
{{eqn | r = \paren {s - 1} \pi^{-s / 2}... | Product Equation for Riemann Zeta Function | https://proofwiki.org/wiki/Product_Equation_for_Riemann_Zeta_Function | https://proofwiki.org/wiki/Product_Equation_for_Riemann_Zeta_Function | [
"Riemann Zeta Function",
"Product Equation for Riemann Zeta Function"
] | [
"Definition:Riemann Zeta Function",
"Definition:Riemann Zeta Function/Zero/Nontrivial",
"Definition:Gamma Function"
] | [
"Definition:Completed Riemann Zeta Function",
"Gamma Difference Equation",
"Definition:Natural Logarithm/Complex",
"Definition:Derivative/Complex Function",
"Completed Riemann Zeta Function has Order One",
"Hadamard Factorization Theorem",
"Definition:Root of Mapping",
"Definition:Riemann Zeta Functio... |
proofwiki-3418 | Zeroes of Functions of Finite Order | Let $\map f z$ be an entire function which satisfies:
:$\map f 0 \ne 0$
:$\cmod {\map f z} \ll \map \exp {\map \alpha {\cmod z} }$
for all $z \in \C$ and some function $\alpha$, where $\ll$ is the order notation.
For $T \ge 1$, let:
:$\map N T = \# \set {\rho \in \C: \map f r = 0, \ \cmod \rho < T}$
where $\#$ denotes ... | Fix $T \ge 1$ and let $\rho_1, \rho_2, \ldots, \rho_n$ be an enumeration of the zeroes of $f$ with modulus less than $T$, counted with multiplicity.
By Jensen's Formula:
:$\ds \frac 1 {2 \pi} \int_0^{2 \pi} \ln \size {\map f {T e^{i \theta} } } \rd \theta = \ln \cmod {\map f 0} + \sum_{k \mathop = 1}^n \paren {\ln T - ... | Let $\map f z$ be an [[Definition:Entire Function|entire function]] which satisfies:
:$\map f 0 \ne 0$
:$\cmod {\map f z} \ll \map \exp {\map \alpha {\cmod z} }$
for all $z \in \C$ and some function $\alpha$, where $\ll$ is the [[Definition:Big-O Notation|order notation]].
For $T \ge 1$, let:
:$\map N T = \# \set {... | Fix $T \ge 1$ and let $\rho_1, \rho_2, \ldots, \rho_n$ be an enumeration of the [[Definition:Zero of Function|zeroes]] of $f$ with modulus less than $T$, counted with [[Definition:Multiplicity (Complex Analysis)|multiplicity]].
By [[Jensen's Formula]]:
:$\ds \frac 1 {2 \pi} \int_0^{2 \pi} \ln \size {\map f {T e^{i \t... | Zeroes of Functions of Finite Order | https://proofwiki.org/wiki/Zeroes_of_Functions_of_Finite_Order | https://proofwiki.org/wiki/Zeroes_of_Functions_of_Finite_Order | [
"Entire Functions"
] | [
"Definition:Entire Function",
"Definition:Big-O Notation",
"Definition:Cardinality"
] | [
"Definition:Root of Mapping",
"Definition:Multiplicity (Complex Analysis)",
"Jensen's Formula",
"Integration by Substitution",
"Jensen's Formula",
"Category:Entire Functions"
] |
proofwiki-3419 | Strongly Locally Compact Space is Weakly Locally Compact | Let $T = \struct {S, \tau}$ be a strongly locally compact space.
Then $T$ is weakly locally compact. | {{Recall|Weakly Locally Compact Space|weakly locally compact space}}
{{:Definition:Weakly Locally Compact Space}}
Let $T = \struct {S, \tau}$ be strongly locally compact.
{{Recall|Strongly Locally Compact Space|strongly locally compact space}}
{{:Definition:Strongly Locally Compact Space/Definition 1}}
Let $x \in S$.
B... | Let $T = \struct {S, \tau}$ be a [[Definition:Strongly Locally Compact Space|strongly locally compact space]].
Then $T$ is [[Definition:Weakly Locally Compact Space|weakly locally compact]]. | {{Recall|Weakly Locally Compact Space|weakly locally compact space}}
{{:Definition:Weakly Locally Compact Space}}
Let $T = \struct {S, \tau}$ be [[Definition:Strongly Locally Compact Space|strongly locally compact]].
{{Recall|Strongly Locally Compact Space|strongly locally compact space}}
{{:Definition:Strongly Local... | Strongly Locally Compact Space is Weakly Locally Compact | https://proofwiki.org/wiki/Strongly_Locally_Compact_Space_is_Weakly_Locally_Compact | https://proofwiki.org/wiki/Strongly_Locally_Compact_Space_is_Weakly_Locally_Compact | [
"Weakly Locally Compact Spaces",
"Strongly Locally Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Strongly Locally Compact Space",
"Definition:Weakly Locally Compact Space"
] | [
"Definition:Strongly Locally Compact Space",
"Definition:Open Set/Topology",
"Definition:Compact Topological Space/Subspace",
"Definition:Closure (Topology)",
"Set is Subset of its Topological Closure",
"Definition:Compact Topological Space/Subspace",
"Definition:Neighborhood (Topology)/Point",
"Defin... |
proofwiki-3420 | Weakly Locally Compact Hausdorff Space is Strongly Locally Compact | Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space.
Let $T$ be weakly locally compact.
Then $T$ is strongly locally compact. | Let $x \in S$.
As $T$ is weakly locally compact, $x$ is contained in a compact neighborhood $N_x$.
As $T$ is a $T_2$ (Hausdorff) space, we can use the result Compact Subspace of Hausdorff Space is Closed.
Thus the interior of $N_x$ has a closure which is compact.
Hence the result, from definition of strongly locally co... | Let $T = \struct {S, \tau}$ be a [[Definition:Hausdorff Space|$T_2$ (Hausdorff) space]].
Let $T$ be [[Definition:Weakly Locally Compact Space|weakly locally compact]].
Then $T$ is [[Definition:Strongly Locally Compact Space|strongly locally compact]]. | Let $x \in S$.
As $T$ is [[Definition:Weakly Locally Compact Space|weakly locally compact]], $x$ is contained in a [[Definition:Compact Topological Subspace|compact]] [[Definition:Neighborhood of Point|neighborhood]] $N_x$.
As $T$ is a [[Definition:Hausdorff Space|$T_2$ (Hausdorff) space]], we can use the result [[Co... | Weakly Locally Compact Hausdorff Space is Strongly Locally Compact | https://proofwiki.org/wiki/Weakly_Locally_Compact_Hausdorff_Space_is_Strongly_Locally_Compact | https://proofwiki.org/wiki/Weakly_Locally_Compact_Hausdorff_Space_is_Strongly_Locally_Compact | [
"Hausdorff Spaces",
"Weakly Locally Compact Spaces",
"Strongly Locally Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:T2 Space",
"Definition:Weakly Locally Compact Space",
"Definition:Strongly Locally Compact Space"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:Compact Topological Space/Subspace",
"Definition:Neighborhood (Topology)/Point",
"Definition:T2 Space",
"Compact Subspace of Hausdorff Space is Closed",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)",
"Definition:Compact Topolog... |
proofwiki-3421 | Weakly Sigma-Locally Compact iff Weakly Locally Compact and Lindelöf | Let $T = \struct {S, \tau}$ be a topological space.
{{TFAE}}
:$(1): \quad T$ is weakly $\sigma$-locally compact
:$(2): \quad T$ is weakly locally compact and Lindelöf | === 1 implies 2 ===
Let $T = \struct {S, \tau}$ be weakly $\sigma$-locally compact.
Then by definition:
:$T$ is $\sigma$-compact
:$T$ is weakly locally compact.
From $\sigma$-Compact Space is Lindelöf Space it follows directly that:
:$T$ is Lindelöf
:$T$ is weakly locally compact.
{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
{{TFAE}}
:$(1): \quad T$ is [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact]]
:$(2): \quad T$ is [[Definition:Weakly Locally Compact Space|weakly locally compact]] and [[Definition:Lindelöf Space|Linde... | === 1 implies 2 ===
Let $T = \struct {S, \tau}$ be [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact]].
Then [[Definition:Weakly Sigma-Locally Compact Space|by definition]]:
:$T$ is [[Definition:Sigma-Compact Space|$\sigma$-compact]]
:$T$ is [[Definition:Weakly Locally Compact Space|wea... | Weakly Sigma-Locally Compact iff Weakly Locally Compact and Lindelöf | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_iff_Weakly_Locally_Compact_and_Lindelöf | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_iff_Weakly_Locally_Compact_and_Lindelöf | [
"Weakly Locally Compact Spaces",
"Weakly Sigma-Locally Compact Spaces",
"Lindelöf Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Topological Space",
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:Weakly Locally Compact Space",
"Definition:Lindelöf Space"
] | [
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:Sigma-Compact Space",
"Definition:Weakly Locally Compact Space",
"Sigma-Compact Space is Lindelöf",
"Definition:Lindelöf Space",
"Definition:Weakly Locally Compact Space",
"Definition:Weakly L... |
proofwiki-3422 | Second-Countable Space is Lindelöf | Let $T = \struct {S, \tau}$ be a topological space which is second-countable.
Then $T$ is also a Lindelöf space. | {{Recall|Lindelöf Space|Lindelöf space}}
{{:Definition:Lindelöf Space}}
Let $T = \struct {S, \tau}$ be a second-countable space.
{{Recall|Second-Countable Space|second-countable space}}
{{:Definition:Second-Countable Space}}
Let $\BB$ be such a countable basis.
Let $\CC$ be an open cover of $T$.
An arbitrary $C \in \CC... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Second-Countable Space|second-countable]].
Then $T$ is also a [[Definition:Lindelöf Space|Lindelöf space]]. | {{Recall|Lindelöf Space|Lindelöf space}}
{{:Definition:Lindelöf Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Second-Countable Space|second-countable space]].
{{Recall|Second-Countable Space|second-countable space}}
{{:Definition:Second-Countable Space}}
Let $\BB$ be such a [[Definition:Countable Basis|count... | Second-Countable Space is Lindelöf | https://proofwiki.org/wiki/Second-Countable_Space_is_Lindelöf | https://proofwiki.org/wiki/Second-Countable_Space_is_Lindelöf | [
"Second-Countable Spaces",
"Lindelöf Spaces"
] | [
"Definition:Topological Space",
"Definition:Second-Countable Space",
"Definition:Lindelöf Space"
] | [
"Definition:Second-Countable Space",
"Definition:Countable Basis",
"Definition:Open Cover",
"Definition:Arbitrary",
"Definition:Set Union",
"Definition:Subset",
"Definition:Countable Set",
"Definition:Arbitrary",
"Definition:Set",
"Definition:Subcover/Countable",
"Definition:Lindelöf Space"
] |
proofwiki-3423 | Separable Space satisfies Countable Chain Condition | Let $T = \struct {S, \tau}$ be a separable topological space.
Then $T$ satisfies the countable chain condition. | In order to demonstrate that $T$ satisfies the '''countable chain condition''', it is sufficient to demonstrate that every disjoint set of open sets of $T$ is countable.
Because $T$ is separable, there exists a subset $\set {y_n : n \in \N}$ of $S$ which is everywhere dense in $S$.
Now consider an indexed family $\fami... | Let $T = \struct {S, \tau}$ be a [[Definition:Separable Space|separable]] [[Definition:Topological Space|topological space]].
Then $T$ satisfies the [[Definition:Countable Chain Condition|countable chain condition]]. | In order to demonstrate that $T$ satisfies the '''[[Definition:Countable Chain Condition|countable chain condition]]''', it is sufficient to demonstrate that every [[Definition:Disjoint Sets|disjoint set]] of [[Definition:Open Set (Topology)|open sets]] of $T$ is [[Definition:Countable Set|countable]].
Because $T$ is... | Separable Space satisfies Countable Chain Condition | https://proofwiki.org/wiki/Separable_Space_satisfies_Countable_Chain_Condition | https://proofwiki.org/wiki/Separable_Space_satisfies_Countable_Chain_Condition | [
"Separable Spaces",
"Countable Chain Condition"
] | [
"Definition:Separable Space",
"Definition:Topological Space",
"Definition:Countable Chain Condition"
] | [
"Definition:Countable Chain Condition",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:Countable Set",
"Definition:Separable Space",
"Definition:Subset",
"Definition:Everywhere Dense",
"Definition:Indexing Set/Family of Sets",
"Definition:Non-Empty Set",
"Definition:Open S... |
proofwiki-3424 | Logarithmic Derivative of Riemann Zeta Function | Let $\zeta$ be the Riemann zeta function:
:$\ds \forall s \in \C: \map \Re s > 1: \map \zeta s = \sum_{n \mathop \ge 1} n^{-s}$
Then for all $s \in \C$ with $\map \Re s > 1$:
:$\ds -\frac {\map {\zeta'} s} {\map \zeta s} = \sum_{n \mathop \ge 1} \map \Lambda n n^{-s}$
where $\Lambda$ is von Mangoldt's function. | By Sum of Reciprocals of Powers as Euler Product:
:$\ds \map \zeta s = \prod_p \frac 1 {1 - p^{-s} }$
where $p$ ranges over the primes.
From Laws of Logarithms:
:$\ds \ln \map \zeta s = - \sum_p \map \ln {1 - p^{-s} }$
{{explain|Is the above equality really true for complex logarithm? I believe no. Probably, we need to... | Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]]:
:$\ds \forall s \in \C: \map \Re s > 1: \map \zeta s = \sum_{n \mathop \ge 1} n^{-s}$
Then for all $s \in \C$ with $\map \Re s > 1$:
:$\ds -\frac {\map {\zeta'} s} {\map \zeta s} = \sum_{n \mathop \ge 1} \map \Lambda n n^{-s}$
where $\L... | By [[Sum of Reciprocals of Powers as Euler Product]]:
:$\ds \map \zeta s = \prod_p \frac 1 {1 - p^{-s} }$
where $p$ ranges over the [[Definition:Prime Number|primes]].
From [[Laws of Logarithms]]:
:$\ds \ln \map \zeta s = - \sum_p \map \ln {1 - p^{-s} }$
{{explain|Is the above equality really true for [[Definition... | Logarithmic Derivative of Riemann Zeta Function | https://proofwiki.org/wiki/Logarithmic_Derivative_of_Riemann_Zeta_Function | https://proofwiki.org/wiki/Logarithmic_Derivative_of_Riemann_Zeta_Function | [
"Riemann Zeta Function"
] | [
"Definition:Riemann Zeta Function",
"Definition:Von Mangoldt Function"
] | [
"Sum of Reciprocals of Powers as Euler Product",
"Definition:Prime Number",
"Laws of Logarithms",
"Definition:Natural Logarithm/Complex",
"Logarithmic Derivative of Infinite Product of Analytic Functions",
"Derivative of Riemann Zeta Function",
"Sum of Infinite Geometric Sequence",
"Definition:Von Man... |
proofwiki-3425 | Riemann Zeta Has No Zeroes With Real Part One | Let $\zeta$ be the Riemann zeta function.
Then for all $t \in \R$:
:$\map \zeta {1 + i t} \ne 0$ | Throughout, the complex variable $s$ is $s = \sigma + i t$.
We have, for $\sigma > 1$,
{{begin-eqn}}
{{eqn | l = -\frac {\map {\zeta'} s} {\map \zeta s}
| r = \sum_{n \mathop \ge 1} \map \Lambda n n^{-s}
| c = Logarithmic Derivative of Riemann Zeta Function
}}
{{eqn | l =
| r = \sum_{n \mathop \ge 1}... | Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
Then for all $t \in \R$:
:$\map \zeta {1 + i t} \ne 0$ | Throughout, the complex variable $s$ is $s = \sigma + i t$.
We have, for $\sigma > 1$,
{{begin-eqn}}
{{eqn | l = -\frac {\map {\zeta'} s} {\map \zeta s}
| r = \sum_{n \mathop \ge 1} \map \Lambda n n^{-s}
| c = [[Logarithmic Derivative of Riemann Zeta Function]]
}}
{{eqn | l =
| r = \sum_{n \mathop ... | Riemann Zeta Has No Zeroes With Real Part One | https://proofwiki.org/wiki/Riemann_Zeta_Has_No_Zeroes_With_Real_Part_One | https://proofwiki.org/wiki/Riemann_Zeta_Has_No_Zeroes_With_Real_Part_One | [
"Riemann Zeta Function"
] | [
"Definition:Riemann Zeta Function"
] | [
"Logarithmic Derivative of Riemann Zeta Function",
"Euler's Formula",
"Poles of Riemann Zeta Function",
"Definition:Order of Pole/Simple Pole",
"Definition:Residue (Complex Analysis)",
"Definition:Order of Zero",
"Definition:Contradiction",
"Category:Riemann Zeta Function"
] |
proofwiki-3426 | Compact Space is Countably Compact | Let $T = \struct {S, \tau}$ be a compact space.
Then $T$ is a countably compact space. | {{Recall|Countably Compact Space|countably compact space}}
{{:Definition:Countably Compact Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a compact space.
{{Recall|Compact Topological Space|compact space}}
{{:Definition:Compact Topological Space/Definition 1}}
So every countable open cover of $S$ has a finite subc... | Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact space]].
Then $T$ is a [[Definition:Countably Compact Space|countably compact space]]. | {{Recall|Countably Compact Space|countably compact space}}
{{:Definition:Countably Compact Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact space]].
{{Recall|Compact Topological Space|compact space}}
{{:Definition:Compact Topological Space/Definition 1}}
So every [... | Compact Space is Countably Compact | https://proofwiki.org/wiki/Compact_Space_is_Countably_Compact | https://proofwiki.org/wiki/Compact_Space_is_Countably_Compact | [
"Compact Topological Spaces",
"Countably Compact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Compact Topological Space",
"Definition:Countably Compact Space"
] | [
"Definition:Compact Topological Space",
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Subcover/Finite",
"Definition:Countably Compact Space/Definition 1",
"Definition:Countably Compact Space"
] |
proofwiki-3427 | Compact Space is Countably Compact | Let $T = \struct {S, \tau}$ be a compact space.
Then $T$ is a countably compact space. | {{Recall|Countably Compact Space|countably compact space|index = 3}}
{{:Definition:Countably Compact Space/Definition 3}}
Let $T = \struct {S, \tau}$ be a sequentially compact topological space.
{{Recall|Sequentially Compact Space|sequentially compact space}}
{{:Definition:Sequentially Compact Space}}
The result follow... | Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact space]].
Then $T$ is a [[Definition:Countably Compact Space|countably compact space]]. | {{Recall|Countably Compact Space|countably compact space|index = 3}}
{{:Definition:Countably Compact Space/Definition 3}}
Let $T = \struct {S, \tau}$ be a [[Definition:Sequentially Compact Space|sequentially compact]] [[Definition:Topological Space|topological space]].
{{Recall|Sequentially Compact Space|sequentially... | Sequentially Compact Space is Countably Compact/Proof 1 | https://proofwiki.org/wiki/Compact_Space_is_Countably_Compact | https://proofwiki.org/wiki/Sequentially_Compact_Space_is_Countably_Compact/Proof_1 | [
"Compact Topological Spaces",
"Countably Compact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Compact Topological Space",
"Definition:Countably Compact Space"
] | [
"Definition:Sequentially Compact Space",
"Definition:Topological Space",
"Limit of Sequence is Accumulation Point"
] |
proofwiki-3428 | Compact Space is Countably Compact | Let $T = \struct {S, \tau}$ be a compact space.
Then $T$ is a countably compact space. | {{Recall|Countably Compact Space|countably compact space|index = 1}}
{{:Definition:Countably Compact Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a sequentially compact topological space.
{{Recall|Sequentially Compact Space|sequentially compact space}}
{{:Definition:Sequentially Compact Space}}
{{AimForCont}} $T... | Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact space]].
Then $T$ is a [[Definition:Countably Compact Space|countably compact space]]. | {{Recall|Countably Compact Space|countably compact space|index = 1}}
{{:Definition:Countably Compact Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a [[Definition:Sequentially Compact Space|sequentially compact]] [[Definition:Topological Space|topological space]].
{{Recall|Sequentially Compact Space|sequentially... | Sequentially Compact Space is Countably Compact/Proof 2 | https://proofwiki.org/wiki/Compact_Space_is_Countably_Compact | https://proofwiki.org/wiki/Sequentially_Compact_Space_is_Countably_Compact/Proof_2 | [
"Compact Topological Spaces",
"Countably Compact Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Compact Topological Space",
"Definition:Countably Compact Space"
] | [
"Definition:Sequentially Compact Space",
"Definition:Topological Space",
"Definition:Countably Compact Space",
"Definition:Cover of Set/Countable",
"Definition:Open Cover",
"Definition:Subcover/Finite",
"Definition:Arbitrary",
"Definition:Sequence/Infinite Sequence",
"Definition:Subsequence",
"Def... |
proofwiki-3429 | Second-Countable Space is Compact iff Countably Compact | Let $T = \struct {S, \tau}$ be a second-countable space.
Then $T$ is compact {{iff}} $T$ is countably compact. | Let $T = \struct {S, \tau}$ be a second-countable space. | Let $T = \struct {S, \tau}$ be a [[Definition:Second-Countable Space|second-countable space]].
Then $T$ is [[Definition:Compact Topological Space|compact]] {{iff}} $T$ is [[Definition:Countably Compact Space|countably compact]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Second-Countable Space|second-countable space]]. | Second-Countable Space is Compact iff Countably Compact | https://proofwiki.org/wiki/Second-Countable_Space_is_Compact_iff_Countably_Compact | https://proofwiki.org/wiki/Second-Countable_Space_is_Compact_iff_Countably_Compact | [
"Second-Countable Spaces",
"Compact Topological Spaces",
"Countably Compact Spaces"
] | [
"Definition:Second-Countable Space",
"Definition:Compact Topological Space",
"Definition:Countably Compact Space"
] | [
"Definition:Second-Countable Space",
"Definition:Second-Countable Space"
] |
proofwiki-3430 | Unsymmetric Functional Equation for Riemann Zeta Function | Let $\zeta$ be the Riemann zeta function.
Let $\Gamma$ be the gamma function.
Then for all $s \in \C$:
:$\map \zeta {1 - s} = 2^{1 - s} \pi^{-s} \map \cos {\dfrac {\pi s} 2} \map \Gamma s \map \zeta s$ | For $s \notin \Z$, we have Euler's Reflection Formula:
:$\map \Gamma s \map \Gamma {1 - s} = \dfrac \pi {\map \sin {\pi s} }$
Replacing $s \mapsto \dfrac {1 + s} 2$ we deduce:
{{begin-eqn}}
{{eqn | l = \map \Gamma {\frac {1 + s} 2} \, \map \Gamma {\frac {1 - s} 2}
| r = \frac \pi {\map \sin {\pi \paren {1 + s} / ... | Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
Let $\Gamma$ be the [[Definition:Gamma Function|gamma function]].
Then for all $s \in \C$:
:$\map \zeta {1 - s} = 2^{1 - s} \pi^{-s} \map \cos {\dfrac {\pi s} 2} \map \Gamma s \map \zeta s$ | For $s \notin \Z$, we have [[Euler's Reflection Formula]]:
:$\map \Gamma s \map \Gamma {1 - s} = \dfrac \pi {\map \sin {\pi s} }$
Replacing $s \mapsto \dfrac {1 + s} 2$ we deduce:
{{begin-eqn}}
{{eqn | l = \map \Gamma {\frac {1 + s} 2} \, \map \Gamma {\frac {1 - s} 2}
| r = \frac \pi {\map \sin {\pi \paren {1 ... | Unsymmetric Functional Equation for Riemann Zeta Function | https://proofwiki.org/wiki/Unsymmetric_Functional_Equation_for_Riemann_Zeta_Function | https://proofwiki.org/wiki/Unsymmetric_Functional_Equation_for_Riemann_Zeta_Function | [
"Riemann Zeta Function",
"Reflection Formulas"
] | [
"Definition:Riemann Zeta Function",
"Definition:Gamma Function"
] | [
"Euler's Reflection Formula",
"Sine and Cosine are Periodic on Reals",
"Legendre's Duplication Formula",
"Functional Equation for Riemann Zeta Function",
"Category:Riemann Zeta Function",
"Category:Reflection Formulas"
] |
proofwiki-3431 | Countably Compact First-Countable Space is Sequentially Compact | A countably compact first-countable topological space is also sequentially compact. | Follows directly from:
: Infinite Sequence in Countably Compact Space has Accumulation Point
: Accumulation Point of Infinite Sequence in First-Countable Space is Subsequential Limit
{{qed}} | A [[Definition:Countably Compact Space|countably compact]] [[Definition:First-Countable Space|first-countable]] [[Definition:Topological Space|topological space]] is also [[Definition:Sequentially Compact Space|sequentially compact]]. | Follows directly from:
: [[Infinite Sequence in Countably Compact Space has Accumulation Point]]
: [[Accumulation Point of Infinite Sequence in First-Countable Space is Subsequential Limit]]
{{qed}} | Countably Compact First-Countable Space is Sequentially Compact/Proof 1 | https://proofwiki.org/wiki/Countably_Compact_First-Countable_Space_is_Sequentially_Compact | https://proofwiki.org/wiki/Countably_Compact_First-Countable_Space_is_Sequentially_Compact/Proof_1 | [
"Countably Compact First-Countable Space is Sequentially Compact",
"Sequentially Compact Spaces",
"Countably Compact Spaces",
"First-Countable Spaces"
] | [
"Definition:Countably Compact Space",
"Definition:First-Countable Space",
"Definition:Topological Space",
"Definition:Sequentially Compact Space"
] | [
"Infinite Sequence in Countably Compact Space has Accumulation Point",
"Accumulation Point of Infinite Sequence in First-Countable Space is Subsequential Limit"
] |
proofwiki-3432 | Countably Compact First-Countable Space is Sequentially Compact | A countably compact first-countable topological space is also sequentially compact. | Let $T = \struct {S, \tau}$ be a countably compact first-countable topological space.
By definition of sequentially compact, it is sufficient to show that every infinite sequence in $S$ has a convergent subsequence.
Let $\sequence {s_n}$ be any sequence in $S$.
By Infinite Sequence in Countably Compact Space has Accumu... | A [[Definition:Countably Compact Space|countably compact]] [[Definition:First-Countable Space|first-countable]] [[Definition:Topological Space|topological space]] is also [[Definition:Sequentially Compact Space|sequentially compact]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Countably Compact Space|countably compact]] [[Definition:First-Countable Space|first-countable]] [[Definition:Topological Space|topological space]].
By definition of [[Definition:Sequentially Compact Space|sequentially compact]], it is sufficient to show that every [[Defin... | Countably Compact First-Countable Space is Sequentially Compact/Proof 2 | https://proofwiki.org/wiki/Countably_Compact_First-Countable_Space_is_Sequentially_Compact | https://proofwiki.org/wiki/Countably_Compact_First-Countable_Space_is_Sequentially_Compact/Proof_2 | [
"Countably Compact First-Countable Space is Sequentially Compact",
"Sequentially Compact Spaces",
"Countably Compact Spaces",
"First-Countable Spaces"
] | [
"Definition:Countably Compact Space",
"Definition:First-Countable Space",
"Definition:Topological Space",
"Definition:Sequentially Compact Space"
] | [
"Definition:Countably Compact Space",
"Definition:First-Countable Space",
"Definition:Topological Space",
"Definition:Sequentially Compact Space",
"Definition:Sequence/Infinite Sequence",
"Definition:Convergent Sequence/Topology",
"Definition:Subsequence",
"Definition:Sequence/Infinite Sequence",
"I... |
proofwiki-3433 | Residue at Simple Pole | Let $f: \C \to \C$ be a function meromorphic on some region, $D$, containing $a$.
Let $f$ have a simple pole at $a$.
Then the residue of $f$ at $a$ is given by:
:$\ds \Res f a = \lim_{z \mathop \to a} \paren {z - a} \map f z$ | By Existence of Laurent Series, there exists a Laurent series:
:$\ds \map f z = \sum_{n \mathop = -\infty}^\infty c_n \paren {z - a}^n$
which is convergent in $D \setminus \set a$, where $\sequence {c_n}$ is a doubly infinite sequence of complex coefficients.
We are given that $f$ has only a simple pole at $a$.
Thus $... | Let $f: \C \to \C$ be a [[Definition:Complex Function|function]] [[Definition:Meromorphic Function|meromorphic]] on some [[Definition:Region (Complex Analysis)|region]], $D$, containing $a$.
Let $f$ have a [[Definition:Simple Pole|simple pole]] at $a$.
Then the [[Definition:Residue (Complex Analysis)|residue]] of $... | By [[Existence of Laurent Series]], there exists a [[Definition:Laurent Series|Laurent series]]:
:$\ds \map f z = \sum_{n \mathop = -\infty}^\infty c_n \paren {z - a}^n$
which is [[Definition:Convergent Series of Numbers|convergent]] in $D \setminus \set a$, where $\sequence {c_n}$ is a [[Definition:Doubly Infinite S... | Residue at Simple Pole | https://proofwiki.org/wiki/Residue_at_Simple_Pole | https://proofwiki.org/wiki/Residue_at_Simple_Pole | [
"Complex Analysis"
] | [
"Definition:Complex Function",
"Definition:Meromorphic Function",
"Definition:Region/Complex",
"Definition:Order of Pole/Simple Pole",
"Definition:Residue (Complex Analysis)"
] | [
"Existence of Laurent Series",
"Definition:Laurent Series",
"Definition:Convergent Series/Number Field",
"Definition:Doubly Infinite Sequence",
"Definition:Complex Number",
"Definition:Coefficient",
"Definition:Order of Pole/Simple Pole",
"Category:Complex Analysis"
] |
proofwiki-3434 | Poles of Riemann Zeta Function | Let $\zeta$ be the Riemann zeta function.
Then $\zeta$ has a simple pole at $s = 1$ with residue $1$, and no other poles. | By Analytic Continuation of Riemann Zeta Function using Mellin Transform of Fractional Part:
:$\ds \map \zeta s = \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x$
is meromorphic for $\map \Re s > 0$, and the integral converges to a finite value for fixed $s$ in this region.
Therefore in this region the ... | Let $\zeta$ be the [[Definition:Riemann Zeta Function|Riemann zeta function]].
Then $\zeta$ has a [[Definition:Simple Pole|simple pole]] at $s = 1$ with [[Definition:Residue (Complex Analysis)|residue]] $1$, and no other [[Definition:Pole (Complex Analysis)|poles]]. | By [[Analytic Continuation of Riemann Zeta Function using Mellin Transform of Fractional Part]]:
:$\ds \map \zeta s = \frac s {s - 1} - s \int_1^\infty \fractpart x x^{-s - 1} \rd x$
is [[Definition:Meromorphic Function|meromorphic]] for $\map \Re s > 0$, and the integral [[Definition:Convergence|converges]] to a fin... | Poles of Riemann Zeta Function | https://proofwiki.org/wiki/Poles_of_Riemann_Zeta_Function | https://proofwiki.org/wiki/Poles_of_Riemann_Zeta_Function | [
"Riemann Zeta Function"
] | [
"Definition:Riemann Zeta Function",
"Definition:Order of Pole/Simple Pole",
"Definition:Residue (Complex Analysis)",
"Definition:Isolated Singularity/Pole"
] | [
"Analytic Continuation of Riemann Zeta Function using Mellin Transform of Fractional Part",
"Definition:Meromorphic Function",
"Definition:Convergence",
"Definition:Region/Plane",
"Definition:Region/Plane",
"Definition:Isolated Singularity/Pole",
"Definition:Residue (Complex Analysis)",
"Residue at Si... |
proofwiki-3435 | Ring of Arithmetic Functions is Ring with Unity | Let $\AA$ be the set of all arithmetic functions.
Let $*$ denote Dirichlet convolution, and $+$ the pointwise sum of functions.
The ring of arithmetic functions $\struct {\AA, +, *}$ is a commutative ring with unity. | By Structure Induced by Abelian Group Operation is Abelian Group, $\struct {\AA, +}$ is an abelian group.
By Properties of Dirichlet Convolution, $*$ is commutative, associative and has a unity.
Therefore $\struct {\AA, +, *}$ is a commutative ring with unity.
{{finish|link to a proof for distributive}}
Category:Arithm... | Let $\AA$ be the set of all [[Definition:Arithmetic Function|arithmetic functions]].
Let $*$ denote [[Definition:Dirichlet Convolution|Dirichlet convolution]], and $+$ the [[Definition:Pointwise Addition|pointwise sum]] of functions.
The [[Definition:Ring of Arithmetic Functions|ring of arithmetic functions]] $\stru... | By [[Structure Induced by Abelian Group Operation is Abelian Group]], $\struct {\AA, +}$ is an [[Definition:Abelian Group|abelian group]].
By [[Properties of Dirichlet Convolution]], $*$ is [[Definition:Commutative Operation|commutative]], [[Definition:Associative Operation|associative]] and has a [[Definition:Unity o... | Ring of Arithmetic Functions is Ring with Unity | https://proofwiki.org/wiki/Ring_of_Arithmetic_Functions_is_Ring_with_Unity | https://proofwiki.org/wiki/Ring_of_Arithmetic_Functions_is_Ring_with_Unity | [
"Arithmetic Functions",
"Examples of Rings"
] | [
"Definition:Arithmetic Function",
"Definition:Dirichlet Convolution",
"Definition:Pointwise Addition",
"Definition:Ring of Arithmetic Functions",
"Definition:Commutative and Unitary Ring"
] | [
"Structure Induced by Abelian Group Operation is Abelian Group",
"Definition:Abelian Group",
"Properties of Dirichlet Convolution",
"Definition:Commutative/Operation",
"Definition:Associative Operation",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Commutative and Unitary Ring",
"Category:A... |
proofwiki-3436 | Units of Ring of Arithmetic Functions | Let $f$ be an arithmetic function.
Then $f$ is a unit in the ring of arithmetic functions {{iff}}:
:$\map f 1 \ne 0$ | Follows immediately from Invertibility of Arithmetic Functions and the definition of ring of arithmetic functions.
{{qed}}
Category:Arithmetic Functions
ms9ywwi9foyyqyxreu2xt3go953vc5z | Let $f$ be an [[Definition:Arithmetic Function|arithmetic function]].
Then $f$ is a [[Definition:Unit of Ring|unit]] in the [[Definition:Ring of Arithmetic Functions|ring of arithmetic functions]] {{iff}}:
:$\map f 1 \ne 0$ | Follows immediately from [[Invertibility of Arithmetic Functions]] and the definition of [[Definition:Ring of Arithmetic Functions|ring of arithmetic functions]].
{{qed}}
[[Category:Arithmetic Functions]]
ms9ywwi9foyyqyxreu2xt3go953vc5z | Units of Ring of Arithmetic Functions | https://proofwiki.org/wiki/Units_of_Ring_of_Arithmetic_Functions | https://proofwiki.org/wiki/Units_of_Ring_of_Arithmetic_Functions | [
"Arithmetic Functions"
] | [
"Definition:Arithmetic Function",
"Definition:Unit of Ring",
"Definition:Ring of Arithmetic Functions"
] | [
"Invertibility of Arithmetic Functions",
"Definition:Ring of Arithmetic Functions",
"Category:Arithmetic Functions"
] |
proofwiki-3437 | Möbius Inversion Formula | Let $f$ and $g$ be arithmetic functions.
Then:
:$(1): \quad \ds \map f n = \sum_{d \mathop \divides n} \map g d$
{{iff}}:
:$(2): \quad \ds \map g n = \sum_{d \mathop \divides n} \map f d \, \map \mu {\frac n d}$
where:
:$d \divides n$ denotes that $d$ is a divisor of $n$
:$\mu$ is the Möbius function. | Let $u$ be the unit arithmetic function and $\iota$ the identity arithmetic function.
Let $*$ denote Dirichlet convolution.
Then equation $(1)$ states that $f = g * u$ and $(2)$ states that $g = f * \mu$.
The proof rests on the following facts:
By Sum of Möbius Function over Divisors:
:$\mu * u = \iota$
By Properties ... | Let $f$ and $g$ be [[Definition:Arithmetic Function|arithmetic functions]].
Then:
:$(1): \quad \ds \map f n = \sum_{d \mathop \divides n} \map g d$
{{iff}}:
:$(2): \quad \ds \map g n = \sum_{d \mathop \divides n} \map f d \, \map \mu {\frac n d}$
where:
:$d \divides n$ denotes that $d$ is a [[Definition:Divisor o... | Let $u$ be the [[Definition:Unit Arithmetic Function|unit arithmetic function]] and $\iota$ the [[Definition:Identity Arithmetic Function|identity arithmetic function]].
Let $*$ denote [[Definition:Dirichlet Convolution|Dirichlet convolution]].
Then equation $(1)$ states that $f = g * u$ and $(2)$ states that $g = f... | Möbius Inversion Formula | https://proofwiki.org/wiki/Möbius_Inversion_Formula | https://proofwiki.org/wiki/Möbius_Inversion_Formula | [
"Möbius Inversion Formula",
"Dirichlet Convolution",
"Möbius Function"
] | [
"Definition:Arithmetic Function",
"Definition:Divisor (Algebra)/Integer",
"Definition:Möbius Function"
] | [
"Definition:Unit Arithmetic Function",
"Definition:Identity Arithmetic Function",
"Definition:Dirichlet Convolution",
"Sum of Möbius Function over Divisors",
"Properties of Dirichlet Convolution",
"Definition:Dirichlet Convolution",
"Definition:Commutative/Operation",
"Definition:Associative Operation... |
proofwiki-3438 | Fully Normal Space is Normal | Let $T = \struct {S, \tau}$ be a fully normal space.
Then $T$ is a normal space. | {{Recall|Fully Normal Space|fully normal space}}
{{:Definition:Fully Normal Space/Definition 2}}
We have that a fully $T_4$ space is also a $T_4$ space.
So:
:$T$ is a $T_4$ Space
:$T$ is a $T_1$ space
which is precisely the definition of a normal space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Fully Normal Space|fully normal space]].
Then $T$ is a [[Definition:Normal Space|normal space]]. | {{Recall|Fully Normal Space|fully normal space}}
{{:Definition:Fully Normal Space/Definition 2}}
We have that a [[Fully T4 Space is T4 Space|fully $T_4$ space is also a $T_4$ space]].
So:
:$T$ is a [[Definition:T4 Space|$T_4$ Space]]
:$T$ is a [[Definition:T1 Space|$T_1$ space]]
which is precisely the definition of a... | Fully Normal Space is Normal | https://proofwiki.org/wiki/Fully_Normal_Space_is_Normal | https://proofwiki.org/wiki/Fully_Normal_Space_is_Normal | [
"Fully Normal Spaces",
"Normal Spaces"
] | [
"Definition:Fully Normal Space",
"Definition:Normal Space"
] | [
"Fully T4 Space is T4",
"Definition:T4 Space",
"Definition:T1 Space",
"Definition:Normal Space"
] |
proofwiki-3439 | Paracompact Space is Countably Paracompact | Let $T = \struct {S, \tau}$ be a paracompact space.
Then $T$ is a countably paracompact space. | {{Recall|Countably Paracompact Space|countably paracompact space}}
{{:Definition:Countably Paracompact Space}}
Let $T = \struct {S, \tau}$ be a paracompact space.
{{Recall|Paracompact Space|paracompact space}}
{{:Definition:Paracompact Space}}
This applies in particular to all countable open covers.
So every countable ... | Let $T = \struct {S, \tau}$ be a [[Definition:Paracompact Space|paracompact space]].
Then $T$ is a [[Definition:Countably Paracompact Space|countably paracompact space]]. | {{Recall|Countably Paracompact Space|countably paracompact space}}
{{:Definition:Countably Paracompact Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Paracompact Space|paracompact space]].
{{Recall|Paracompact Space|paracompact space}}
{{:Definition:Paracompact Space}}
This applies in particular to all [[Defi... | Paracompact Space is Countably Paracompact | https://proofwiki.org/wiki/Paracompact_Space_is_Countably_Paracompact | https://proofwiki.org/wiki/Paracompact_Space_is_Countably_Paracompact | [
"Paracompact Spaces",
"Countably Paracompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Paracompact Space",
"Definition:Countably Paracompact Space"
] | [
"Definition:Paracompact Space",
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Open Refinement",
"Definition:Locally Finite Cover",
"Definition:Countably Paracompact Space"
] |
proofwiki-3440 | Metacompact Space is Countably Metacompact | Let $T = \struct {S, \tau}$ be a metacompact space.
Then $T$ is a countably metacompact space. | From the definition, $T$ is metacompact space {{iff}} every open cover of $S$ has an open refinement which is point finite.
This also applies to all countable open covers.
So every countable open cover of $S$ has an open refinement which is point finite.
This is precisely the definition for a countably metacompact spac... | Let $T = \struct {S, \tau}$ be a [[Definition:Metacompact Space|metacompact space]].
Then $T$ is a [[Definition:Countably Metacompact Space|countably metacompact space]]. | From the definition, $T$ is [[Definition:Metacompact Space|metacompact space]] {{iff}} every [[Definition:Open Cover|open cover]] of $S$ has an [[Definition:Open Refinement|open refinement]] which is [[Definition:Point Finite Cover|point finite]].
This also applies to all [[Definition:Countable Set|countable]] [[Defin... | Metacompact Space is Countably Metacompact | https://proofwiki.org/wiki/Metacompact_Space_is_Countably_Metacompact | https://proofwiki.org/wiki/Metacompact_Space_is_Countably_Metacompact | [
"Metacompact Spaces",
"Countably Metacompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Metacompact Space",
"Definition:Countably Metacompact Space"
] | [
"Definition:Metacompact Space",
"Definition:Open Cover",
"Definition:Open Refinement",
"Definition:Point Finite Cover",
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Open Refinement",
"Definition:Point Finite Cover",
"Defini... |
proofwiki-3441 | Fully T4 Space is T4 | Let $T = \struct {S, \tau}$ be a fully $T_4$ space.
Then $T$ is a $T_4$ space. | {{Recall|T4 Space|$T_4$ space}}
{{:Definition:T4 Space/Definition 1}}
Consider the open cover $\UU = \set {\relcomp S A, \relcomp S B}$.
Since $T$ is fully $T_4$, there exists a barycentric refinement $\VV$ of $\UU$.
Define $\VV_A$ and $\VV_B$ by:
:$\VV_A := \set {V \in \VV: A \cap V \ne \O}$
:$\VV_B := \set {V \in \VV... | Let $T = \struct {S, \tau}$ be a [[Definition:Fully T4 Space|fully $T_4$ space]].
Then $T$ is a [[Definition:T4 Space|$T_4$ space]]. | {{Recall|T4 Space|$T_4$ space}}
{{:Definition:T4 Space/Definition 1}}
Consider the [[Definition:Open Cover|open cover]] $\UU = \set {\relcomp S A, \relcomp S B}$.
Since $T$ is [[Definition:Fully T4 Space|fully $T_4$]], there exists a [[Definition:Barycentric Refinement|barycentric refinement]] $\VV$ of $\UU$.
Define... | Fully T4 Space is T4 | https://proofwiki.org/wiki/Fully_T4_Space_is_T4 | https://proofwiki.org/wiki/Fully_T4_Space_is_T4 | [
"Fully T4 Spaces",
"T4 Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Fully T4 Space",
"Definition:T4 Space"
] | [
"Definition:Open Cover",
"Definition:Fully T4 Space",
"Definition:Barycentric Refinement",
"Definition:Open Cover",
"Definition:Star (Topology)",
"Definition:Fully T4 Space",
"Definition:Contradiction",
"Definition:T4 Space"
] |
proofwiki-3442 | Fully Normal Space is Paracompact | Let $T = \struct {S, \tau}$ be a fully normal space.
Then $T$ is paracompact. | {{Recall|Paracompact Space|paracompact}}
{{:Definition:Paracompact Space}}
Let $T = \struct {S, \tau}$ be a fully normal space.
{{Recall|Fully Normal Space|fully normal}}
{{:Definition:Fully Normal Space/Definition 2}}
{{Recall|Fully T4 Space|fully $T_4$}}
{{:Definition:Fully T4 Space}}
Let $\UU$ be an open cover for $... | Let $T = \struct {S, \tau}$ be a [[Definition:Fully Normal Space|fully normal space]].
Then $T$ is [[Definition:Paracompact Space|paracompact]]. | {{Recall|Paracompact Space|paracompact}}
{{:Definition:Paracompact Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Fully Normal Space|fully normal space]].
{{Recall|Fully Normal Space|fully normal}}
{{:Definition:Fully Normal Space/Definition 2}}
{{Recall|Fully T4 Space|fully $T_4$}}
{{:Definition:Fully T4 Spa... | Fully Normal Space is Paracompact/Proof 1 | https://proofwiki.org/wiki/Fully_Normal_Space_is_Paracompact | https://proofwiki.org/wiki/Fully_Normal_Space_is_Paracompact/Proof_1 | [
"Fully Normal Space is Paracompact",
"Fully Normal Spaces",
"Paracompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Fully Normal Space",
"Definition:Paracompact Space"
] | [
"Definition:Fully Normal Space",
"Definition:Open Cover",
"Definition:Barycentric Refinement",
"Definition:Cover of Set",
"Fully T4 Space is Paracompact"
] |
proofwiki-3443 | Fully Normal Space is Paracompact | Let $T = \struct {S, \tau}$ be a fully normal space.
Then $T$ is paracompact. | Let $T = \struct {S, \tau}$ be a fully normal space.
{{Recall|Fully Normal Space|fully normal}}
{{:Definition:Fully Normal Space/Definition 2}}
Thus $T$ is a fully $T_4$ space.
The result follows from Fully $T_4$ Space is Paracompact. | Let $T = \struct {S, \tau}$ be a [[Definition:Fully Normal Space|fully normal space]].
Then $T$ is [[Definition:Paracompact Space|paracompact]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Fully Normal Space|fully normal space]].
{{Recall|Fully Normal Space|fully normal}}
{{:Definition:Fully Normal Space/Definition 2}}
Thus $T$ is a [[Definition:Fully T4 Space|fully $T_4$ space]].
The result follows from [[Fully T4 Space is Paracompact|Fully $T_4$ Space is... | Fully Normal Space is Paracompact/Proof 2 | https://proofwiki.org/wiki/Fully_Normal_Space_is_Paracompact | https://proofwiki.org/wiki/Fully_Normal_Space_is_Paracompact/Proof_2 | [
"Fully Normal Space is Paracompact",
"Fully Normal Spaces",
"Paracompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Fully Normal Space",
"Definition:Paracompact Space"
] | [
"Definition:Fully Normal Space",
"Definition:Fully T4 Space",
"Fully T4 Space is Paracompact"
] |
proofwiki-3444 | Subcover is Refinement of Cover | Let $S$ be a set.
Let $\CC$ be a cover for $S$.
Let $\VV$ be a subcover of $\CC$.
Then $\VV$ is a refinement of $\CC$. | From definition of subcover:
:$\VV \subseteq \CC$
That is, every element of $\VV$ is an element of $\CC$.
From definition of subset, every element of $\VV$ is the subset of some element of $\CC$.
This is precisely the definition of refinement.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\CC$ be a [[Definition:Cover of Set|cover]] for $S$.
Let $\VV$ be a [[Definition:Subcover|subcover]] of $\CC$.
Then $\VV$ is a [[Definition:Refinement of Cover|refinement]] of $\CC$. | From definition of [[Definition:Subcover|subcover]]:
:$\VV \subseteq \CC$
That is, every [[Definition:Element|element]] of $\VV$ is an [[Definition:Element|element]] of $\CC$.
From definition of [[Definition:Subset|subset]], every [[Definition:Element|element]] of $\VV$ is the [[Definition:Subset|subset]] of some [[D... | Subcover is Refinement of Cover | https://proofwiki.org/wiki/Subcover_is_Refinement_of_Cover | https://proofwiki.org/wiki/Subcover_is_Refinement_of_Cover | [
"Subcovers",
"Refinements of Covers",
"Covers"
] | [
"Definition:Set",
"Definition:Cover of Set",
"Definition:Subcover",
"Definition:Refinement of Cover"
] | [
"Definition:Subcover",
"Definition:Element",
"Definition:Element",
"Definition:Subset",
"Definition:Element",
"Definition:Subset",
"Definition:Element",
"Definition:Refinement of Cover"
] |
proofwiki-3445 | Compact Space is Paracompact | Let $T = \struct {S, \tau}$ be a compact space.
Then $T$ is paracompact. | {{Recall|Compact Topological Space|compact space}}
{{:Definition:Compact Topological Space}}
From Subcover is Refinement of Cover, it follows that every open cover of $S$ has an open refinement which is locally finite.
This is precisely the definition of paracompact.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact space]].
Then $T$ is [[Definition:Paracompact Space|paracompact]]. | {{Recall|Compact Topological Space|compact space}}
{{:Definition:Compact Topological Space}}
From [[Subcover is Refinement of Cover]], it follows that every [[Definition:Open Cover|open cover]] of $S$ has an [[Definition:Open Refinement|open refinement]] which is [[Definition:Locally Finite Cover|locally finite]].
Th... | Compact Space is Paracompact/Proof 1 | https://proofwiki.org/wiki/Compact_Space_is_Paracompact | https://proofwiki.org/wiki/Compact_Space_is_Paracompact/Proof_1 | [
"Compact Space is Paracompact",
"Compact Topological Spaces",
"Paracompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Compact Topological Space",
"Definition:Paracompact Space"
] | [
"Subcover is Refinement of Cover",
"Definition:Open Cover",
"Definition:Open Refinement",
"Definition:Locally Finite Cover",
"Definition:Paracompact Space"
] |
proofwiki-3446 | Compact Space is Paracompact | Let $T = \struct {S, \tau}$ be a compact space.
Then $T$ is paracompact. | Follows directly from:
{{begin-itemize}}
{{item||Compact Space is Strongly Paracompact}}
{{item||Strongly Paracompact Space is Paracompact}}.
{{end-itemize}}
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact space]].
Then $T$ is [[Definition:Paracompact Space|paracompact]]. | Follows directly from:
{{begin-itemize}}
{{item||[[Compact Space is Strongly Paracompact]]}}
{{item||[[Strongly Paracompact Space is Paracompact]]}}.
{{end-itemize}}
{{qed}} | Compact Space is Paracompact/Proof 2 | https://proofwiki.org/wiki/Compact_Space_is_Paracompact | https://proofwiki.org/wiki/Compact_Space_is_Paracompact/Proof_2 | [
"Compact Space is Paracompact",
"Compact Topological Spaces",
"Paracompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Compact Topological Space",
"Definition:Paracompact Space"
] | [
"Compact Space is Strongly Paracompact",
"Strongly Paracompact Space is Paracompact"
] |
proofwiki-3447 | Paracompact Space is Metacompact | Let $T = \struct {S, \tau}$ be a paracompact space.
Then $T$ is also metacompact. | {{Recall|Metacompact Space|metacompact space}}
{{:Definition:Metacompact Space}}
Let $T = \struct {S, \tau}$ be a paracompact space.
{{Recall|Paracompact Space|paracompact space}}
{{:Definition:Paracompact Space}}
Consider an arbitrary open cover $\CC$ of $S$.
Then {{apriori}} there exists an open refinement $\UU$ of $... | Let $T = \struct {S, \tau}$ be a [[Definition:Paracompact Space|paracompact space]].
Then $T$ is also [[Definition:Metacompact Space|metacompact]]. | {{Recall|Metacompact Space|metacompact space}}
{{:Definition:Metacompact Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Paracompact Space|paracompact space]].
{{Recall|Paracompact Space|paracompact space}}
{{:Definition:Paracompact Space}}
Consider an [[Definition:Arbitrary|arbitrary]] [[Definition:Open Cover... | Paracompact Space is Metacompact | https://proofwiki.org/wiki/Paracompact_Space_is_Metacompact | https://proofwiki.org/wiki/Paracompact_Space_is_Metacompact | [
"Paracompact Spaces",
"Metacompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Paracompact Space",
"Definition:Metacompact Space"
] | [
"Definition:Paracompact Space",
"Definition:Arbitrary",
"Definition:Open Cover",
"Definition:Open Refinement",
"Definition:Locally Finite Cover",
"Locally Finite Refinement is Point Finite",
"Definition:Point Finite Cover",
"Definition:Metacompact Space"
] |
proofwiki-3448 | Countably Paracompact Space is Countably Metacompact | Let $T = \struct {S, \tau}$ be a countably paracompact space.
Then $T$ is countably metacompact. | {{Recall|Countably Paracompact Space|countably paracompact space}}
{{:Definition:Countably Paracompact Space}}
{{Recall|Countably Metacompact Space|countably metacompact space}}
{{:Definition:Countably Metacompact Space}}
The result follows from Locally Finite Refinement is Point Finite.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Countably Paracompact Space|countably paracompact space]].
Then $T$ is [[Definition:Countably Metacompact Space|countably metacompact]]. | {{Recall|Countably Paracompact Space|countably paracompact space}}
{{:Definition:Countably Paracompact Space}}
{{Recall|Countably Metacompact Space|countably metacompact space}}
{{:Definition:Countably Metacompact Space}}
The result follows from [[Locally Finite Refinement is Point Finite]].
{{qed}} | Countably Paracompact Space is Countably Metacompact | https://proofwiki.org/wiki/Countably_Paracompact_Space_is_Countably_Metacompact | https://proofwiki.org/wiki/Countably_Paracompact_Space_is_Countably_Metacompact | [
"Countably Paracompact Spaces",
"Countably Metacompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Countably Paracompact Space",
"Definition:Countably Metacompact Space"
] | [
"Locally Finite Refinement is Point Finite"
] |
proofwiki-3449 | Countably Compact Space is Countably Paracompact | Let $T = \struct {S, \tau}$ be a countably compact space.
Then $T$ is countably paracompact. | {{Recall|Countably Paracompact Space|countably paracompact space}}
{{:Definition:Countably Paracompact Space}}
Let $T = \struct {S, \tau}$ be a countably compact space.
{{Recall|Countably Compact Space|countably compact space}}
{{:Definition:Countably Compact Space/Definition 1}}
Let $\CC$ be an arbitrary countable ope... | Let $T = \struct {S, \tau}$ be a [[Definition:Countably Compact Space|countably compact space]].
Then $T$ is [[Definition:Countably Paracompact Space|countably paracompact]]. | {{Recall|Countably Paracompact Space|countably paracompact space}}
{{:Definition:Countably Paracompact Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Countably Compact Space|countably compact space]].
{{Recall|Countably Compact Space|countably compact space}}
{{:Definition:Countably Compact Space/Definition 1}... | Countably Compact Space is Countably Paracompact | https://proofwiki.org/wiki/Countably_Compact_Space_is_Countably_Paracompact | https://proofwiki.org/wiki/Countably_Compact_Space_is_Countably_Paracompact | [
"Countably Compact Spaces",
"Countably Paracompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Countably Compact Space",
"Definition:Countably Paracompact Space"
] | [
"Definition:Countably Compact Space",
"Definition:Arbitrary",
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Subcover/Finite",
"Subcover is Refinement of Cover",
"Definition:Refinement of Cover",
"Subcover of Open Cover is Open",
"Definition:Open Cover",
"Finite Cover is Locally ... |
proofwiki-3450 | Metacompact Countably Compact Space is Compact | Let $T = \struct {S, \tau}$ be a countably compact space which is also metacompact.
Then $T$ is compact. | Let $T = \struct {S, \tau}$ be a countably compact space which is also metacompact.
{{Recall|Countably Compact Space|countably compact space}}
{{:Definition:Countably Compact Space/Definition 1}}
{{Recall|Metacompact Space|metacompact}}
{{:Definition:Metacompact Space}}
Let $\UU_\alpha$ be an open cover of $S$.
Then le... | Let $T = \struct {S, \tau}$ be a [[Definition:Countably Compact Space|countably compact space]] which is also [[Definition:Metacompact Space|metacompact]].
Then $T$ is [[Definition:Compact Topological Space|compact]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Countably Compact Space|countably compact space]] which is also [[Definition:Metacompact Space|metacompact]].
{{Recall|Countably Compact Space|countably compact space}}
{{:Definition:Countably Compact Space/Definition 1}}
{{Recall|Metacompact Space|metacompact}}
{{:Defini... | Metacompact Countably Compact Space is Compact | https://proofwiki.org/wiki/Metacompact_Countably_Compact_Space_is_Compact | https://proofwiki.org/wiki/Metacompact_Countably_Compact_Space_is_Compact | [
"Metacompact Spaces",
"Compact Topological Spaces",
"Countably Compact Spaces"
] | [
"Definition:Countably Compact Space",
"Definition:Metacompact Space",
"Definition:Compact Topological Space"
] | [
"Definition:Countably Compact Space",
"Definition:Metacompact Space",
"Definition:Open Cover",
"Definition:Open Refinement",
"Definition:Point Finite Cover",
"Definition:Metacompact Space",
"Definition:Element",
"Definition:Finite Set",
"Definition:Element",
"Definition:Point Finite Cover",
"Def... |
proofwiki-3451 | Countably Metacompact Lindelöf Space is Metacompact | Let $T = \struct {S, \tau}$ be a Lindelöf space which is also countably metacompact.
Then $T$ is metacompact. | By the definitions:
:If $T = \struct {S, \tau}$ is a Lindelöf space then every open cover of $S$ has a countable subcover.
:If $T = \struct {S, \tau}$ is a countably metacompact space then every countable open cover of $S$ has an open refinement which is point finite.
It follows trivially that every open cover of $S$ h... | Let $T = \struct {S, \tau}$ be a [[Definition:Lindelöf Space|Lindelöf space]] which is also [[Definition:Countably Metacompact Space|countably metacompact]].
Then $T$ is [[Definition:Metacompact Space|metacompact]]. | By the definitions:
:If $T = \struct {S, \tau}$ is a [[Definition:Lindelöf Space|Lindelöf space]] then every [[Definition:Open Cover|open cover]] of $S$ has a [[Definition:Countable Subcover|countable subcover]].
:If $T = \struct {S, \tau}$ is a [[Definition:Countably Metacompact Space|countably metacompact space]] th... | Countably Metacompact Lindelöf Space is Metacompact | https://proofwiki.org/wiki/Countably_Metacompact_Lindelöf_Space_is_Metacompact | https://proofwiki.org/wiki/Countably_Metacompact_Lindelöf_Space_is_Metacompact | [
"Countably Metacompact Spaces",
"Metacompact Spaces",
"Lindelöf Spaces"
] | [
"Definition:Lindelöf Space",
"Definition:Countably Metacompact Space",
"Definition:Metacompact Space"
] | [
"Definition:Lindelöf Space",
"Definition:Open Cover",
"Definition:Subcover/Countable",
"Definition:Countably Metacompact Space",
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Open Refinement",
"Definition:Point Finite Cover",
"Definition:Open Cover",
"Definition:Open Refinement"... |
proofwiki-3452 | Countably Paracompact Lindelöf Space is Paracompact | Let $T = \struct {S, \tau}$ be a Lindelöf space which is also countably paracompact.
Then $T$ is paracompact. | By the definitions:
:If $T = \struct {S, \tau}$ is a Lindelöf space then every open cover of $S$ has a countable subcover.
:If $T = \struct {S, \tau}$ is a countably paracompact space then every countable open cover of $S$ has an open refinement which is locally finite.
It follows trivially that every open cover of $S$... | Let $T = \struct {S, \tau}$ be a [[Definition:Lindelöf Space|Lindelöf space]] which is also [[Definition:Countably Paracompact Space|countably paracompact]].
Then $T$ is [[Definition:Paracompact Space|paracompact]]. | By the definitions:
:If $T = \struct {S, \tau}$ is a [[Definition:Lindelöf Space|Lindelöf space]] then every [[Definition:Open Cover|open cover]] of $S$ has a [[Definition:Countable Subcover|countable subcover]].
:If $T = \struct {S, \tau}$ is a [[Definition:Countably Paracompact Space|countably paracompact space]] th... | Countably Paracompact Lindelöf Space is Paracompact | https://proofwiki.org/wiki/Countably_Paracompact_Lindelöf_Space_is_Paracompact | https://proofwiki.org/wiki/Countably_Paracompact_Lindelöf_Space_is_Paracompact | [
"Countably Paracompact Spaces",
"Paracompact Spaces",
"Lindelöf Spaces"
] | [
"Definition:Lindelöf Space",
"Definition:Countably Paracompact Space",
"Definition:Paracompact Space"
] | [
"Definition:Lindelöf Space",
"Definition:Open Cover",
"Definition:Subcover/Countable",
"Definition:Countably Paracompact Space",
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Open Refinement",
"Definition:Locally Finite Cover",
"Definition:Open Cover",
"Definition:Open Refinemen... |
proofwiki-3453 | Dirichlet's Approximation Theorem | Let $\alpha, x \in \R$.
Then there exist integers $a, q$ such that:
:$\gcd \set {a, q} = 1$
:$1 \le q \le x$
and:
:$\size {\alpha - \dfrac a q} \le \dfrac 1 {q x}$ | It is sufficient to find $a, q$ not necessarily coprime.
Once such an $a, q$ have been found, then a coprime pair can be found by dividing numerator and denominator of $\dfrac a q$ by the GCD of $a$ and $q$.
Let $X = \floor x$ be the integer part of $x$.
Let $\alpha_q = \alpha q - \floor {\alpha q} \in \hointr 0 1$ for... | Let $\alpha, x \in \R$.
Then there exist [[Definition:Integer|integers]] $a, q$ such that:
:$\gcd \set {a, q} = 1$
:$1 \le q \le x$
and:
:$\size {\alpha - \dfrac a q} \le \dfrac 1 {q x}$ | It is sufficient to find $a, q$ not necessarily [[Definition:Coprime Integers|coprime]].
Once such an $a, q$ have been found, then a [[Definition:Coprime Integers|coprime]] pair can be found by dividing [[Definition:Numerator|numerator]] and [[Definition:Denominator|denominator]] of $\dfrac a q$ by the [[Definition:Gr... | Dirichlet's Approximation Theorem | https://proofwiki.org/wiki/Dirichlet's_Approximation_Theorem | https://proofwiki.org/wiki/Dirichlet's_Approximation_Theorem | [
"Diophantine Approximation",
"Diophantine Equations"
] | [
"Definition:Integer"
] | [
"Definition:Coprime/Integers",
"Definition:Coprime/Integers",
"Definition:Fraction/Numerator",
"Definition:Fraction/Denominator",
"Definition:Greatest Common Divisor/Integers",
"Definition:Floor Function",
"Definition:Pairwise Disjoint",
"Definition:Real Interval",
"Dirichlet's Box Principle/Corolla... |
proofwiki-3454 | Convergence of Dirichlet Series with Bounded Partial Sums | Let $\sequence {a_n}_{n \mathop \in \N}$ be a sequence in $\C$.
Suppose that there exists $B > 0$ such that for all $n, m \in \N$:
:$\ds \size {\sum_{k \mathop = m}^n a_n} \le B$
Then the Dirichlet series:
:$\ds \map f s = \sum_{n \mathop \ge 1} a_n n^{-s}$
converges locally uniformly to an analytic function on $\map \... | By Exponential is Entire, the partial sums:
:$\ds \map {f_N} s = \sum_{n \mathop = 1}^N a_n n^{-s}$
are analytic.
So by Uniform Limit of Analytic Functions is Analytic it is sufficient to show locally uniform convergence.
For $0 < A < \pi / 2$, $\delta > 0$ we let:
:$D_{A, \delta} = \set {s \in \C : -A < \map \arg s < ... | Let $\sequence {a_n}_{n \mathop \in \N}$ be a [[Definition:Sequence|sequence]] in $\C$.
Suppose that there exists $B > 0$ such that for all $n, m \in \N$:
:$\ds \size {\sum_{k \mathop = m}^n a_n} \le B$
Then the [[Definition:Dirichlet Series|Dirichlet series]]:
:$\ds \map f s = \sum_{n \mathop \ge 1} a_n n^{-s}$
[[D... | By [[Exponential is Entire]], the [[Definition:Partial Sum|partial sums]]:
:$\ds \map {f_N} s = \sum_{n \mathop = 1}^N a_n n^{-s}$
are [[Definition:Analytic Function|analytic]].
So by [[Uniform Limit of Analytic Functions is Analytic]] it is sufficient to show [[Definition:Locally Uniform Convergence|locally uniform... | Convergence of Dirichlet Series with Bounded Partial Sums | https://proofwiki.org/wiki/Convergence_of_Dirichlet_Series_with_Bounded_Partial_Sums | https://proofwiki.org/wiki/Convergence_of_Dirichlet_Series_with_Bounded_Partial_Sums | [
"Analytic Number Theory"
] | [
"Definition:Sequence",
"Definition:Dirichlet Series",
"Definition:Locally Uniform Convergence",
"Definition:Analytic Function"
] | [
"Exponential is Entire",
"Definition:Series/Sequence of Partial Sums",
"Definition:Analytic Function",
"Uniform Limit of Analytic Functions is Analytic",
"Definition:Locally Uniform Convergence",
"Definition:Locally Uniform Convergence",
"Abel's Lemma/Formulation 2",
"Definition:Bounded Uniformly",
... |
proofwiki-3455 | Convergence of Dirichlet Series with Bounded Coefficients | Let $\sequence {a_n}_{n \mathop \in \N}$ be a bounded sequence in $\C$.
Then the Dirichlet series:
:$\ds \map f s = \sum_{n \mathop \ge 1} a_n n^{-s}$
converges absolutely and locally uniformly to an analytic function on $\map \Re s > 1$. | By Exponential is Entire, the partial sums:
:$\ds \map {f_N} s = \sum_{n \mathop = 1}^N a_n n^{-s}$
are analytic.
So by Uniform Limit of Analytic Functions is Analytic it is sufficient to show locally uniform convergence.
Let $B$ be a bound for the $a_n$:
:$\forall n \in \N: \size {a_n} \le B$
Let $D$ be any open subse... | Let $\sequence {a_n}_{n \mathop \in \N}$ be a [[Definition:Bounded Sequence|bounded sequence]] in $\C$.
Then the [[Definition:Dirichlet Series|Dirichlet series]]:
:$\ds \map f s = \sum_{n \mathop \ge 1} a_n n^{-s}$
[[Definition:Absolutely Convergent Series|converges absolutely]] and [[Definition:Locally Uniform Conver... | By [[Exponential is Entire]], the [[Definition:Partial Sum|partial sums]]:
:$\ds \map {f_N} s = \sum_{n \mathop = 1}^N a_n n^{-s}$
are [[Definition:Analytic Function|analytic]].
So by [[Uniform Limit of Analytic Functions is Analytic]] it is sufficient to show [[Definition:Locally Uniform Convergence|locally uniform... | Convergence of Dirichlet Series with Bounded Coefficients | https://proofwiki.org/wiki/Convergence_of_Dirichlet_Series_with_Bounded_Coefficients | https://proofwiki.org/wiki/Convergence_of_Dirichlet_Series_with_Bounded_Coefficients | [
"Analytic Number Theory"
] | [
"Definition:Bounded Sequence",
"Definition:Dirichlet Series",
"Definition:Absolutely Convergent Series",
"Definition:Locally Uniform Convergence",
"Definition:Analytic Function"
] | [
"Exponential is Entire",
"Definition:Series/Sequence of Partial Sums",
"Definition:Analytic Function",
"Uniform Limit of Analytic Functions is Analytic",
"Definition:Locally Uniform Convergence",
"Definition:Bound of Sequence",
"Definition:Open Set/Complex Analysis",
"Convergence of P-Series/Real",
... |
proofwiki-3456 | Dirichlet L-Function from Trivial Character | Let $\chi_0$ be the trivial Dirichlet character modulo $q$.
{{explain|Trivial character we got, Dirichlet character we got, we still need a page for trivial Dirichlet character. There exists on another page a link to Definition:Dirichlet Character/Trivial Character which ought to be straightforward to construct.}}
Let ... | By definition:
:$\map {\chi_0} a = \begin{cases} 1 & : \gcd \set {a, q} = 1 \\ 0 & : \text{otherwise} \end{cases}$
{{explain|Worth specifying the domain of $a$ for added clarity here.}}
Therefore:
{{begin-eqn}}
{{eqn | l = \map L {s, \chi_0}
| r = \prod_p \paren {1 - \map \chi p p^{-s} }^{-1}
| c = {{Defof|... | Let $\chi_0$ be the [[Definition:Trivial Dirichlet Character|trivial]] [[Definition:Dirichlet Character|Dirichlet character]] modulo $q$.
{{explain|Trivial character we got, Dirichlet character we got, we still need a page for trivial Dirichlet character. There exists on another page a link to [[Definition:Dirichlet C... | By definition:
:$\map {\chi_0} a = \begin{cases} 1 & : \gcd \set {a, q} = 1 \\ 0 & : \text{otherwise} \end{cases}$
{{explain|Worth specifying the domain of $a$ for added clarity here.}}
Therefore:
{{begin-eqn}}
{{eqn | l = \map L {s, \chi_0}
| r = \prod_p \paren {1 - \map \chi p p^{-s} }^{-1}
| c = {{De... | Dirichlet L-Function from Trivial Character | https://proofwiki.org/wiki/Dirichlet_L-Function_from_Trivial_Character | https://proofwiki.org/wiki/Dirichlet_L-Function_from_Trivial_Character | [
"Analytic Number Theory"
] | [
"Definition:Trivial Dirichlet Character",
"Definition:Dirichlet Character",
"Definition:Dirichlet Character/Trivial Character",
"Definition:Dirichlet L-function",
"Definition:Riemann Zeta Function",
"Definition:Divisor (Algebra)/Integer"
] | [
"Definition:Prime Number",
"Fundamental Theorem of Arithmetic",
"Fundamental Theorem of Arithmetic",
"Category:Analytic Number Theory"
] |
proofwiki-3457 | Separable Metacompact Space is Lindelöf | Let $T = \struct {S, \tau}$ be a separable topological space which is also metacompact.
Then $T$ is a Lindelöf space. | {{Recall|Separable Space|separable space}}
{{:Definition:Separable Space}}
{{Recall|Metacompact Space|metacompact space}}
{{:Definition:Metacompact Space}}
{{Recall|Lindelöf Space|Lindelöf space}}
{{:Definition:Lindelöf Space}}
Having established the definitions, we proceed.
Let $T = \struct {S, \tau}$ be separable and... | Let $T = \struct {S, \tau}$ be a [[Definition:Separable Space|separable topological space]] which is also [[Definition:Metacompact Space|metacompact]].
Then $T$ is a [[Definition:Lindelöf Space|Lindelöf space]]. | {{Recall|Separable Space|separable space}}
{{:Definition:Separable Space}}
{{Recall|Metacompact Space|metacompact space}}
{{:Definition:Metacompact Space}}
{{Recall|Lindelöf Space|Lindelöf space}}
{{:Definition:Lindelöf Space}}
Having established the definitions, we proceed.
Let $T = \struct {S, \tau}$ be [[Definit... | Separable Metacompact Space is Lindelöf/Proof 1 | https://proofwiki.org/wiki/Separable_Metacompact_Space_is_Lindelöf | https://proofwiki.org/wiki/Separable_Metacompact_Space_is_Lindelöf/Proof_1 | [
"Separable Metacompact Space is Lindelöf",
"Separable Spaces",
"Metacompact Spaces",
"Lindelöf Spaces"
] | [
"Definition:Separable Space",
"Definition:Metacompact Space",
"Definition:Lindelöf Space"
] | [
"Definition:Separable Space",
"Definition:Metacompact Space",
"Definition:Open Cover",
"Definition:Subcover/Countable",
"Definition:Metacompact Space",
"Definition:Open Refinement",
"Definition:Point Finite Cover",
"Definition:Subcover/Countable",
"Definition:Uncountable/Set",
"Definition:Separabl... |
proofwiki-3458 | Separable Metacompact Space is Lindelöf | Let $T = \struct {S, \tau}$ be a separable topological space which is also metacompact.
Then $T$ is a Lindelöf space. | {{Recall|Metacompact Space|metacompact space}}
{{:Definition:Metacompact Space}}
{{Recall|Lindelöf Space|Lindelöf space}}
{{:Definition:Lindelöf Space}}
Having established the definitions, we proceed.
Let $\UU$ be an open cover of $S$.
Let $\VV$ be a point finite open refinement of $\UU$.
By Point Finite Set of Open Se... | Let $T = \struct {S, \tau}$ be a [[Definition:Separable Space|separable topological space]] which is also [[Definition:Metacompact Space|metacompact]].
Then $T$ is a [[Definition:Lindelöf Space|Lindelöf space]]. | {{Recall|Metacompact Space|metacompact space}}
{{:Definition:Metacompact Space}}
{{Recall|Lindelöf Space|Lindelöf space}}
{{:Definition:Lindelöf Space}}
Having established the definitions, we proceed.
Let $\UU$ be an [[Definition:Open Cover|open cover]] of $S$.
Let $\VV$ be a [[Definition:Point Finite Cover|point f... | Separable Metacompact Space is Lindelöf/Proof 2 | https://proofwiki.org/wiki/Separable_Metacompact_Space_is_Lindelöf | https://proofwiki.org/wiki/Separable_Metacompact_Space_is_Lindelöf/Proof_2 | [
"Separable Metacompact Space is Lindelöf",
"Separable Spaces",
"Metacompact Spaces",
"Lindelöf Spaces"
] | [
"Definition:Separable Space",
"Definition:Metacompact Space",
"Definition:Lindelöf Space"
] | [
"Definition:Open Cover",
"Definition:Point Finite Cover",
"Definition:Open Refinement",
"Point Finite Set of Open Sets in Separable Space is Countable",
"Definition:Countable Set",
"Definition:Mapping",
"Image of Countable Set under Mapping is Countable",
"Definition:Image (Set Theory)/Mapping/Mapping... |
proofwiki-3459 | Functional Equation for Dirichlet L-Functions | Let $\chi$ be a primitive Dirichlet character to the modulus $q \geq 1$.
Let $\map \Lambda {s, \chi}$ be the completed $L$-function for $\chi$.
Let $\map \tau \chi$ denote the Gaussian sum.
Then for all $s \in \C$:
:$\map \Lambda {s, \chi} = i^{-\kappa} \dfrac {\map \tau \chi} {\sqrt q} \map \Lambda {1 - s, \overline \... | {{proof wanted}}
Category:Analytic Number Theory
6b4oru2c2fsvpefnpvjo41xwl64m5up | Let $\chi$ be a [[Definition:Primitive Dirichlet Character|primitive Dirichlet character]] to the modulus $q \geq 1$.
Let $\map \Lambda {s, \chi}$ be the [[Definition:Completed Dirichlet L-Function|completed $L$-function]] for $\chi$.
Let $\map \tau \chi$ denote the [[Definition:Gauss Sum|Gaussian sum]].
Then for a... | {{proof wanted}}
[[Category:Analytic Number Theory]]
6b4oru2c2fsvpefnpvjo41xwl64m5up | Functional Equation for Dirichlet L-Functions | https://proofwiki.org/wiki/Functional_Equation_for_Dirichlet_L-Functions | https://proofwiki.org/wiki/Functional_Equation_for_Dirichlet_L-Functions | [
"Analytic Number Theory"
] | [
"Definition:Dirichlet Character/Primitive Character",
"Definition:Completed Dirichlet L-Function",
"Definition:Gauss Sum"
] | [
"Category:Analytic Number Theory"
] |
proofwiki-3460 | Orthogonality Relations for Characters | Let $G$ be a finite abelian group with identity $e$.
Let $G^*$ be the dual group of characters $\chi : G \to \C_{\ne 0}$.
Let $\chi_0$ be the trivial character on $G$.
Let $\psi: G \to \C_{\ne 0}$ be any character.
Let $y \in G$ be arbitrary.
Then:
:$\ds \sum_{x \mathop \in G} \map \psi x = \begin {cases}
\order G & : ... | If $\psi = \chi_0$, then it is straightforward that:
{{begin-eqn}}
{{eqn | l = \sum_{x \mathop \in G} \map \psi x
| r = \sum_{x \mathop \in G} \map {\chi_0} x
| c = {{hypothesis}}
}}
{{eqn | r = \sum_{x \mathop \in G} 1
| c = {{Defof|Trivial Character}}
}}
{{eqn | r = \order G
| c =
}}
{{end-eq... | Let $G$ be a [[Definition:Finite Group|finite]] [[Definition:Abelian Group|abelian group]] with [[Definition:Identity Element|identity]] $e$.
Let $G^*$ be the dual group of [[Definition:Character (Number Theory)|characters]] $\chi : G \to \C_{\ne 0}$.
Let $\chi_0$ be the [[Definition:Trivial Character|trivial charact... | If $\psi = \chi_0$, then it is straightforward that:
{{begin-eqn}}
{{eqn | l = \sum_{x \mathop \in G} \map \psi x
| r = \sum_{x \mathop \in G} \map {\chi_0} x
| c = {{hypothesis}}
}}
{{eqn | r = \sum_{x \mathop \in G} 1
| c = {{Defof|Trivial Character}}
}}
{{eqn | r = \order G
| c =
}}
{{end-e... | Orthogonality Relations for Characters | https://proofwiki.org/wiki/Orthogonality_Relations_for_Characters | https://proofwiki.org/wiki/Orthogonality_Relations_for_Characters | [
"Analytic Number Theory"
] | [
"Definition:Finite Group",
"Definition:Abelian Group",
"Definition:Identity (Abstract Algebra)/Two-Sided Identity",
"Definition:Character (Number Theory)",
"Definition:Trivial Character",
"Definition:Character (Number Theory)"
] | [
"Definition:Summation",
"Definition:Character (Number Theory)",
"Definition:Group Homomorphism",
"Category:Analytic Number Theory"
] |
proofwiki-3461 | Analytic Continuation of Dirichlet L-Function | Let $\chi : G := \paren {\Z / q \Z}^\times \to \C^\times$ be a Dirichlet character modulo $q$.
{{explain|$\C^\times$}}
Let $\map L {s, \chi}$ be the Dirichlet $L$-function for $\chi$.
Let $\chi$ be the trivial character.
Then $\map L {s, \chi}$ has an analytic continuation to $\C$ except for a simple pole at $s = 1$.
L... | Let $\chi$ be the trivial character.
Then by Dirichlet L-Function from Trivial Character:
:$\ds \map L {s, \chi} = \map \zeta s \cdot \prod_{p \mathop \divides q} \paren {1 - p^{-s} }$
where $\divides$ denotes divisibility.
Also, by Poles of Riemann Zeta Function, $\zeta$ is analytic on $\C$ except for a simple pole at... | Let $\chi : G := \paren {\Z / q \Z}^\times \to \C^\times$ be a [[Definition:Dirichlet Character|Dirichlet character]] modulo $q$.
{{explain|$\C^\times$}}
Let $\map L {s, \chi}$ be the [[Definition:Dirichlet L-function|Dirichlet $L$-function]] for $\chi$.
Let $\chi$ be the [[Definition:Trivial Dirichlet Character|tr... | Let $\chi$ be the [[Definition:Trivial Dirichlet Character|trivial character]].
Then by [[Dirichlet L-Function from Trivial Character]]:
:$\ds \map L {s, \chi} = \map \zeta s \cdot \prod_{p \mathop \divides q} \paren {1 - p^{-s} }$
where $\divides$ denotes [[Definition:Divisor of Integer|divisibility]].
Also, by [[... | Analytic Continuation of Dirichlet L-Function | https://proofwiki.org/wiki/Analytic_Continuation_of_Dirichlet_L-Function | https://proofwiki.org/wiki/Analytic_Continuation_of_Dirichlet_L-Function | [
"Analytic Number Theory"
] | [
"Definition:Dirichlet Character",
"Definition:Dirichlet L-function",
"Definition:Trivial Dirichlet Character",
"Definition:Order of Pole/Simple Pole"
] | [
"Definition:Trivial Dirichlet Character",
"Dirichlet L-Function from Trivial Character",
"Definition:Divisor (Algebra)/Integer",
"Poles of Riemann Zeta Function",
"Definition:Trivial Dirichlet Character",
"Orthogonality Relations for Characters",
"Definition:Periodic Function",
"Definition:Coprime/Int... |
proofwiki-3462 | L-Function does not Vanish at One | Let $\psi$ be a non-trivial Dirichlet character modulo $q$.
Let $\map L {s, \chi}$ be the Dirichlet $L$-function associated to $\chi$.
Then $\map L {1, \chi} \ne 0$. | Let $G^*$ be the group of characters modulo $q$.
{{explain|"group of characters modulo $q$" - we have the definition for character, and we have (somewhere) the definition of the ring of integers modulo $q$, but not this entity.}}
Let $\ds \map {\zeta_q} s = \prod_{\chi \mathop \in G^*} \map L {s, \chi} R$.
Among the fa... | Let $\psi$ be a non-[[Definition:Trivial Dirichlet Character|trivial]] [[Definition:Dirichlet Character|Dirichlet character]] modulo $q$.
Let $\map L {s, \chi}$ be the [[Definition:Dirichlet L-function|Dirichlet $L$-function associated to $\chi$]].
Then $\map L {1, \chi} \ne 0$. | Let $G^*$ be the [[Definition:Group|group]] of characters modulo $q$.
{{explain|"[[Definition:Group|group]] of characters modulo $q$" - we have the definition for character, and we have (somewhere) the definition of the ring of integers modulo $q$, but not this entity.}}
Let $\ds \map {\zeta_q} s = \prod_{\chi \mathop... | L-Function does not Vanish at One | https://proofwiki.org/wiki/L-Function_does_not_Vanish_at_One | https://proofwiki.org/wiki/L-Function_does_not_Vanish_at_One | [
"Analytic Number Theory"
] | [
"Definition:Trivial Dirichlet Character",
"Definition:Dirichlet Character",
"Definition:Dirichlet L-function"
] | [
"Definition:Group",
"Definition:Group",
"Analytic Continuation of Dirichlet L-Function",
"Definition:Order of Pole/Simple Pole",
"Definition:Isolated Singularity/Pole",
"Analytic Continuation of Dirichlet L-Function",
"Definition:Prime Number",
"Definition:Euler Product",
"Definition:Absolutely Conv... |
proofwiki-3463 | Logarithm of Dirichlet L-Functions | Let $\chi$ be a Dirichlet character modulo $q$.
The Dirichlet series:
:$\map f s = \ds \sum_{n \mathop \ge 1} \sum_p \frac {\map \chi p^n} {n p^{n s} }$
converges absolutely to an analytic function, where $p$ ranges over the primes.
Moreover, $\map f s$ defines a branch of $\ln \map L {s, \chi}$. | By Convergence of Dirichlet Series with Bounded Coefficients, $\map f s$ converges absolutely on $\map \Re s > 1$ to an analytic function.
For fixed $s \in \set {\map \Re s > 1}$:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop \ge 1} \sum_p \frac {\map \chi p^n} {n p^{n s} }
| r = \sum_p \paren {\frac {\map \chi p} {p... | Let $\chi$ be a [[Definition:Dirichlet Character|Dirichlet character]] modulo $q$.
The [[Definition:Dirichlet Series|Dirichlet series]]:
:$\map f s = \ds \sum_{n \mathop \ge 1} \sum_p \frac {\map \chi p^n} {n p^{n s} }$
[[Definition:Absolutely Convergent Series|converges absolutely]] to an [[Definition:Analytic Func... | By [[Convergence of Dirichlet Series with Bounded Coefficients]], $\map f s$ [[Definition:Absolutely Convergent Series|converges absolutely]] on $\map \Re s > 1$ to an [[Definition:Analytic Function|analytic function]].
For fixed $s \in \set {\map \Re s > 1}$:
{{begin-eqn}}
{{eqn | l = \sum_{n \mathop \ge 1} \sum_p \... | Logarithm of Dirichlet L-Functions | https://proofwiki.org/wiki/Logarithm_of_Dirichlet_L-Functions | https://proofwiki.org/wiki/Logarithm_of_Dirichlet_L-Functions | [
"Analytic Number Theory"
] | [
"Definition:Dirichlet Character",
"Definition:Dirichlet Series",
"Definition:Absolutely Convergent Series",
"Definition:Analytic Function",
"Definition:Prime Number",
"Definition:Branch (Complex Analysis)"
] | [
"Convergence of Dirichlet Series with Bounded Coefficients",
"Definition:Absolutely Convergent Series",
"Definition:Analytic Function",
"Manipulation of Absolutely Convergent Series",
"Taylor Series of Logarithm",
"Logarithm is Continuous",
"Category:Analytic Number Theory"
] |
proofwiki-3464 | Dirichlet's Theorem on Arithmetic Sequences | Let $a, q$ be coprime integers.
Let $\PP_{a, q}$ be the set of primes $p$ such that $p \equiv a \pmod q$.
Then $\PP_{a, q}$ has Dirichlet density:
:$\map \phi q^{-1}$
where $\phi$ is Euler's phi function.
In particular, $\PP_{a, q}$ is infinite. | === Lemma 1 ===
{{:Dirichlet's Theorem on Arithmetic Sequences/Lemma 1}}{{qed|lemma}}
Define:
:$\eta_{a, q} : n \mapsto \begin {cases} 1 & : n \equiv a \pmod q \\ 0 & : \text {otherwise} \end {cases}$ | Let $a, q$ be [[Definition:Coprime Integers|coprime integers]].
Let $\PP_{a, q}$ be the [[Definition:Set|set]] of [[Definition:Prime Number|primes]] $p$ such that $p \equiv a \pmod q$.
Then $\PP_{a, q}$ has [[Definition:Dirichlet Density|Dirichlet density]]:
:$\map \phi q^{-1}$
where $\phi$ is [[Definition:Euler Phi... | === [[Dirichlet's Theorem on Arithmetic Sequences/Lemma 1|Lemma 1]] ===
{{:Dirichlet's Theorem on Arithmetic Sequences/Lemma 1}}{{qed|lemma}}
Define:
:$\eta_{a, q} : n \mapsto \begin {cases} 1 & : n \equiv a \pmod q \\ 0 & : \text {otherwise} \end {cases}$ | Dirichlet's Theorem on Arithmetic Sequences | https://proofwiki.org/wiki/Dirichlet's_Theorem_on_Arithmetic_Sequences | https://proofwiki.org/wiki/Dirichlet's_Theorem_on_Arithmetic_Sequences | [
"Dirichlet's Theorem on Arithmetic Sequences",
"Arithmetic Sequences",
"Prime Numbers",
"Analytic Number Theory"
] | [
"Definition:Coprime/Integers",
"Definition:Set",
"Definition:Prime Number",
"Definition:Dirichlet Density",
"Definition:Euler Phi Function",
"Definition:Infinite Set"
] | [
"Dirichlet's Theorem on Arithmetic Sequences/Lemma 1",
"Dirichlet's Theorem on Arithmetic Sequences/Lemma 1"
] |
proofwiki-3465 | Discrete Fourier Transform on Abelian Group | Let $G$ be a finite abelian group.
Let $G^*$ be the dual group of characters $G \to \C^\times$.
Let $\eta: G \to \C$ be a mapping from $G$ to the set of complex numbers.
Then for all $x \in G$:
:$\ds \map \eta x = \frac 1 {\map \phi q} \sum_{\chi \mathop \in G^*} \innerprod \eta \chi_G \map \chi x$
where:
:$\ds \innerp... | {{begin-eqn}}
{{eqn | l = \frac 1 {\map \phi q} \sum_{\chi \mathop \in G^*} \innerprod \eta \chi_G \map \chi y
| r = \frac 1 {\map \phi q} \sum_{\chi \mathop \in G^*} \sum_{x \mathop \in G} \map \eta x \map {\overline \chi} x \map \chi y
| c =
}}
{{eqn | r = \frac 1 {\map \phi q} \sum_{x \mathop \in G} \map... | Let $G$ be a [[Definition:Finite Group|finite]] [[Definition:Abelian Group|abelian group]].
Let $G^*$ be the dual group of [[Definition:Character (Number Theory)|characters]] $G \to \C^\times$.
Let $\eta: G \to \C$ be a [[Definition:Mapping|mapping]] from $G$ to the set of [[Definition:Complex Number|complex numbers]... | {{begin-eqn}}
{{eqn | l = \frac 1 {\map \phi q} \sum_{\chi \mathop \in G^*} \innerprod \eta \chi_G \map \chi y
| r = \frac 1 {\map \phi q} \sum_{\chi \mathop \in G^*} \sum_{x \mathop \in G} \map \eta x \map {\overline \chi} x \map \chi y
| c =
}}
{{eqn | r = \frac 1 {\map \phi q} \sum_{x \mathop \in G} \map... | Discrete Fourier Transform on Abelian Group | https://proofwiki.org/wiki/Discrete_Fourier_Transform_on_Abelian_Group | https://proofwiki.org/wiki/Discrete_Fourier_Transform_on_Abelian_Group | [
"Fourier Analysis",
"Representation Theory"
] | [
"Definition:Finite Group",
"Definition:Abelian Group",
"Definition:Character (Number Theory)",
"Definition:Mapping",
"Definition:Complex Number"
] | [
"Orthogonality Relations for Characters",
"Category:Fourier Analysis",
"Category:Representation Theory"
] |
proofwiki-3466 | Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods | Let $T = \struct {S, \tau}$ be a Hausdorff space.
Let $V_1$ and $V_2$ be compact sets in $T$ which are disjoint:
:$V_1 \cap V_2 = \O$
Then $V_1$ and $V_2$ have disjoint neighborhoods. | === Lemma ===
{{:Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods/Lemma}}
Let $\FF$ be the set of all ordered pairs $\tuple {Z, W}$ such that:
:$Z, W \in \tau$
:$V_1 \subseteq Z$
:$Z \cap W = \O$
By the {{Lemma|Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods}}, $\Img \FF$ covers... | Let $T = \struct {S, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $V_1$ and $V_2$ be [[Definition:Compact Topological Space|compact sets]] in $T$ which are [[Definition:Disjoint Sets|disjoint]]:
:$V_1 \cap V_2 = \O$
Then $V_1$ and $V_2$ have [[Definition:Disjoint Sets|disjoint]] [[Definition:Neigh... | === [[Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods/Lemma|Lemma]] ===
{{:Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods/Lemma}}
Let $\FF$ be the [[Definition:Set|set]] of all [[Definition:Ordered Pair|ordered pairs]] $\tuple {Z, W}$ such that:
:$Z, W \in \tau$
:$V_1 \subset... | Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods/Proof 1 | https://proofwiki.org/wiki/Disjoint_Compact_Sets_in_Hausdorff_Space_have_Disjoint_Neighborhoods | https://proofwiki.org/wiki/Disjoint_Compact_Sets_in_Hausdorff_Space_have_Disjoint_Neighborhoods/Proof_1 | [
"Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods",
"Hausdorff Spaces",
"Compact Topological Spaces"
] | [
"Definition:T2 Space",
"Definition:Compact Topological Space",
"Definition:Disjoint Sets",
"Definition:Disjoint Sets",
"Definition:Neighborhood (Topology)/Set"
] | [
"Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods/Lemma",
"Definition:Set",
"Definition:Ordered Pair",
"Definition:Cover of Set",
"Definition:Compact Topological Space",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Cover of Set/Finite",
"Definition:Topology",
"Defi... |
proofwiki-3467 | Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods | Let $T = \struct {S, \tau}$ be a Hausdorff space.
Let $V_1$ and $V_2$ be compact sets in $T$ which are disjoint:
:$V_1 \cap V_2 = \O$
Then $V_1$ and $V_2$ have disjoint neighborhoods. | We first suppose that $V_1 = \set x$ and $x \not \in V_2$.
Since $T$ is Hausdorff:
:for each $y \in V_2$ there exists an open neighborhood $O^1_y$ of $x$ and an open neighborhood $O^2_y$ of $y$ such that $O^1_y \cap O^2_y = \O$.
Hence we have:
:$\ds V_2 \subseteq \bigcup_{y \mathop \in V_2} O_y^2$
Since $V_2$ is comp... | Let $T = \struct {S, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff space]].
Let $V_1$ and $V_2$ be [[Definition:Compact Topological Space|compact sets]] in $T$ which are [[Definition:Disjoint Sets|disjoint]]:
:$V_1 \cap V_2 = \O$
Then $V_1$ and $V_2$ have [[Definition:Disjoint Sets|disjoint]] [[Definition:Neigh... | We first suppose that $V_1 = \set x$ and $x \not \in V_2$.
Since $T$ is [[Definition:Hausdorff Space|Hausdorff]]:
:for each $y \in V_2$ there exists an [[Definition:Open Neighborhood|open neighborhood]] $O^1_y$ of $x$ and an [[Definition:Open Neighborhood|open neighborhood]] $O^2_y$ of $y$ such that $O^1_y \cap O^2_y... | Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods/Proof 2 | https://proofwiki.org/wiki/Disjoint_Compact_Sets_in_Hausdorff_Space_have_Disjoint_Neighborhoods | https://proofwiki.org/wiki/Disjoint_Compact_Sets_in_Hausdorff_Space_have_Disjoint_Neighborhoods/Proof_2 | [
"Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods",
"Hausdorff Spaces",
"Compact Topological Spaces"
] | [
"Definition:T2 Space",
"Definition:Compact Topological Space",
"Definition:Disjoint Sets",
"Definition:Disjoint Sets",
"Definition:Neighborhood (Topology)/Set"
] | [
"Definition:T2 Space",
"Definition:Open Neighborhood",
"Definition:Open Neighborhood",
"Definition:Compact Topological Space",
"Definition:Open Set/Topology",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Compact Topological Space",
"Defini... |
proofwiki-3468 | Compact Subsets of T3 Spaces | Let $T = \struct {S, \tau}$ be a $T_3$ space.
Let $A \subseteq S$ be compact in $T$.
Then for each $U \in \tau$ such that $A \subseteq U$:
:$\exists V \in \tau: A \subseteq V \subseteq V^- \subseteq U$
where $V^-$ denotes the closure of $V$. | Let $A \subseteq S$ be compact in $T$.
Let $U \in \tau$ such that $A \subseteq U$.
Since $T$ is $T_3$:
:Each open set contains a closed neighborhood around each of its points:
::$\forall x \in U: \exists N_x: \relcomp S {N_x} \in \tau: \exists V_x \in \tau: x \in V_x \subseteq N_x \subseteq U$
Note that $\set {V_x: x \... | Let $T = \struct {S, \tau}$ be a [[Definition:T3 Space|$T_3$ space]].
Let $A \subseteq S$ be [[Definition:Compact Topological Space|compact]] in $T$.
Then for each $U \in \tau$ such that $A \subseteq U$:
:$\exists V \in \tau: A \subseteq V \subseteq V^- \subseteq U$
where $V^-$ denotes the [[Definition:Closure (Topo... | Let $A \subseteq S$ be [[Definition:Compact Topological Space|compact]] in $T$.
Let $U \in \tau$ such that $A \subseteq U$.
Since $T$ is [[Definition:T3 Space/Definition 2|$T_3$]]:
:Each [[Definition:Open Set (Topology)|open set]] contains a [[Definition:Closed Neighborhood|closed neighborhood]] around each of its ... | Compact Subsets of T3 Spaces | https://proofwiki.org/wiki/Compact_Subsets_of_T3_Spaces | https://proofwiki.org/wiki/Compact_Subsets_of_T3_Spaces | [
"T3 Spaces",
"Compact Topological Spaces"
] | [
"Definition:T3 Space",
"Definition:Compact Topological Space",
"Definition:Closure (Topology)"
] | [
"Definition:Compact Topological Space",
"Definition:T3 Space/Definition 2",
"Definition:Open Set/Topology",
"Definition:Closed Neighborhood",
"Definition:Open Cover",
"Definition:Compact Topological Space",
"Definition:Subcover/Finite",
"Definition:Finite Set",
"Definition:Subset",
"Definition:Ope... |
proofwiki-3469 | Compact Hausdorff Space is T4 | Let $T = \struct {S, \tau}$ be a compact $T_2$ (Hausdorff) space.
Then $T$ is a $T_4$ space. | {{Recall|T4 Space|$T_4$ space|index = 1}}
{{:Definition:T4 Space/Definition 1}}
We have that a Compact Subspace of Hausdorff Space is Closed.
We also have that a Closed Subspace of Compact Space is Compact.
We also have that Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods.
But by the definition abo... | Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact]] [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
Then $T$ is a [[Definition:T4 Space|$T_4$ space]]. | {{Recall|T4 Space|$T_4$ space|index = 1}}
{{:Definition:T4 Space/Definition 1}}
We have that a [[Compact Subspace of Hausdorff Space is Closed]].
We also have that a [[Closed Subspace of Compact Space is Compact]].
We also have that [[Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods]].
But by th... | Compact Hausdorff Space is T4 | https://proofwiki.org/wiki/Compact_Hausdorff_Space_is_T4 | https://proofwiki.org/wiki/Compact_Hausdorff_Space_is_T4 | [
"T4 Spaces",
"Hausdorff Spaces",
"Compact Topological Spaces"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:T4 Space"
] | [
"Compact Subspace of Hausdorff Space is Closed",
"Closed Subspace of Compact Space is Compact",
"Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods",
"Definition:T4 Space",
"Definition:Disjoint Sets",
"Definition:Closed Set/Topology",
"Definition:Separated by Neighborhoods"
] |
proofwiki-3470 | Compact Hausdorff Topology is Minimal Hausdorff | Let $T = \struct {S, \tau}$ be a Hausdorff space which is compact.
Then $\tau$ is the minimal subset of the power set $\powerset S$ such that $T$ is a Hausdorff space. | {{AimForCont}} there exists a topology $\tau'$ on $S$ such that:
:$\tau' \subseteq \tau$ but $\tau' \ne \tau$
:$\tau'$ is a Hausdorff space.
From Equivalence of Definitions of Finer Topology:
:the identity mapping $I_S: \struct {S, \tau} \to \struct {S, \tau'}$ is continuous.
Let $A \in \tau$.
Then $S \setminus A \subs... | Let $T = \struct {S, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff space]] which is [[Definition:Compact Topological Space|compact]].
Then $\tau$ is the [[Definition:Minimal Set|minimal]] [[Definition:Subset|subset]] of the [[Definition:Power Set|power set]] $\powerset S$ such that $T$ is a [[Definition:Hausdorff... | {{AimForCont}} there exists a [[Definition:Topology|topology]] $\tau'$ on $S$ such that:
:$\tau' \subseteq \tau$ but $\tau' \ne \tau$
:$\tau'$ is a [[Definition:Hausdorff Space|Hausdorff space]].
From [[Equivalence of Definitions of Finer Topology]]:
:the [[Definition:Identity Mapping|identity mapping]] $I_S: \struct ... | Compact Hausdorff Topology is Minimal Hausdorff | https://proofwiki.org/wiki/Compact_Hausdorff_Topology_is_Minimal_Hausdorff | https://proofwiki.org/wiki/Compact_Hausdorff_Topology_is_Minimal_Hausdorff | [
"Hausdorff Spaces",
"Compact Topological Spaces"
] | [
"Definition:T2 Space",
"Definition:Compact Topological Space",
"Definition:Minimal/Set",
"Definition:Subset",
"Definition:Power Set",
"Definition:T2 Space"
] | [
"Definition:Topology",
"Definition:T2 Space",
"Equivalence of Definitions of Finer Topology",
"Definition:Identity Mapping",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Closed Set/Topology",
"Closed Subspace of Compact Space is Compact",
"Definition:Compact Topological Space",
... |
proofwiki-3471 | Compact Hausdorff Topology is Maximally Compact | Let $T = \struct {S, \tau}$ be a Hausdorff space which is compact.
Then $\tau$ is maximally compact. | Let $\tau'$ be a topology on $S$ such that $\tau \subseteq \tau'$ but that $\tau \ne \tau'$.
Consider the identity mapping $I_S: \struct {S, \tau'} \to \struct {S, \tau}$.
From Separation Properties Preserved in Subspace, $I_S$ is a continuous bijection from a Hausdorff space to a compact Hausdorff space.
{{AimForCont}... | Let $T = \struct {S, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff space]] which is [[Definition:Compact Topological Space|compact]].
Then $\tau$ is [[Definition:Maximal Set|maximally]] [[Definition:Compact Topological Space|compact]]. | Let $\tau'$ be a [[Definition:Topology|topology]] on $S$ such that $\tau \subseteq \tau'$ but that $\tau \ne \tau'$.
Consider the [[Definition:Identity Mapping|identity mapping]] $I_S: \struct {S, \tau'} \to \struct {S, \tau}$.
From [[Separation Properties Preserved in Subspace]], $I_S$ is a [[Definition:Everywhere C... | Compact Hausdorff Topology is Maximally Compact | https://proofwiki.org/wiki/Compact_Hausdorff_Topology_is_Maximally_Compact | https://proofwiki.org/wiki/Compact_Hausdorff_Topology_is_Maximally_Compact | [
"Hausdorff Spaces",
"Compact Topological Spaces"
] | [
"Definition:T2 Space",
"Definition:Compact Topological Space",
"Definition:Maximal/Set",
"Definition:Compact Topological Space"
] | [
"Definition:Topology",
"Definition:Identity Mapping",
"Separation Properties Preserved in Subspace",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Bijection",
"Definition:T2 Space",
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Compact Topological Sp... |
proofwiki-3472 | Maximum Cardinality of Separable Hausdorff Space | Let $T = \struct {S, \tau}$ be a Hausdorff space which is separable.
Then $S$ can have a cardinality no greater than $2^{2^{\aleph_0} }$. | Let $D$ be an everywhere dense subset of $S$ which is countable, as is guaranteed as $T$ is separable.
Consider the mapping $\Phi: S \to 2^{\powerset D}$ defined as:
:$\forall x \in S: \map {\map \Phi x} A = 1 \iff A = D \cap U_x$ for some neighborhood $U_x$ of $x$
{{explain|It is not clear in Steen & Seeabch what is m... | Let $T = \struct {S, \tau}$ be a [[Definition:Hausdorff Space|Hausdorff space]] which is [[Definition:Separable Space|separable]].
Then $S$ can have a [[Definition:Cardinality|cardinality]] no greater than $2^{2^{\aleph_0} }$. | Let $D$ be an [[Definition:Everywhere Dense|everywhere dense]] [[Definition:Subset|subset]] of $S$ which is [[Definition:Countable Set|countable]], as is guaranteed as $T$ is [[Definition:Separable Space|separable]].
Consider the [[Definition:Mapping|mapping]] $\Phi: S \to 2^{\powerset D}$ defined as:
:$\forall x \in ... | Maximum Cardinality of Separable Hausdorff Space | https://proofwiki.org/wiki/Maximum_Cardinality_of_Separable_Hausdorff_Space | https://proofwiki.org/wiki/Maximum_Cardinality_of_Separable_Hausdorff_Space | [
"Hausdorff Spaces",
"Separable Spaces"
] | [
"Definition:T2 Space",
"Definition:Separable Space",
"Definition:Cardinality"
] | [
"Definition:Everywhere Dense",
"Definition:Subset",
"Definition:Countable Set",
"Definition:Separable Space",
"Definition:Mapping",
"Definition:Neighborhood (Topology)/Point",
"Definition:T2 Space",
"Definition:Injection"
] |
proofwiki-3473 | Cluster Point of Ultrafilter is Unique | Let $S$ be a set.
Let $\FF$ be an ultrafilter on $S$.
Let $x \in S$ be a cluster point of $\FF$.
Then there is no point $y \in S: y \ne x$ such that $y$ is also a cluster point of $\FF$. | Let $x, y$ both be cluster points of an ultrafilter $\FF$ on a set $S$.
Then by definition:
:$\forall U \in \FF: \set {x, y} \subseteq U$
But then we can create a finer filter $\FF \cup \set {\set x}$ on $S$.
So $\FF$ is not an ultrafilter.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\FF$ be an [[Definition:Ultrafilter on Set|ultrafilter]] on $S$.
Let $x \in S$ be a [[Definition:Cluster Point of Filter|cluster point]] of $\FF$.
Then there is no point $y \in S: y \ne x$ such that $y$ is also a [[Definition:Cluster Point of Filter|cluster point]] of $\FF$... | Let $x, y$ both be [[Definition:Cluster Point of Filter|cluster points]] of an [[Definition:Ultrafilter on Set|ultrafilter]] $\FF$ on a [[Definition:Set|set]] $S$.
Then by definition:
:$\forall U \in \FF: \set {x, y} \subseteq U$
But then we can create a [[Definition:Finer Filter on Set|finer filter]] $\FF \cup \set ... | Cluster Point of Ultrafilter is Unique | https://proofwiki.org/wiki/Cluster_Point_of_Ultrafilter_is_Unique | https://proofwiki.org/wiki/Cluster_Point_of_Ultrafilter_is_Unique | [
"Cluster Points of Filters",
"Ultrafilters on Sets"
] | [
"Definition:Set",
"Definition:Ultrafilter on Set",
"Definition:Cluster Point of Filter",
"Definition:Cluster Point of Filter"
] | [
"Definition:Cluster Point of Filter",
"Definition:Ultrafilter on Set",
"Definition:Set",
"Definition:Finer Filter on Set",
"Definition:Ultrafilter on Set"
] |
proofwiki-3474 | Sequence of Implications of Separation Axioms | Let $P_1$ and $P_2$ be separation axioms and let:
:$P_1 \implies P_2$
mean:
:If a topological space $T$ satsifies separation axiom $P_1$, then $T$ also satisfies separation axiom $P_2$.
Then the following sequence of separation axioms holds:
{|
|-
| align="center" | ||
| align="center" | ||
| align="center" | Perfectly... | The relevant justifications are listed as follows:
* Perfectly Normal implies Perfectly $T_4$ by definition.
* Perfectly $T_4$ Space is $T_5$
* Perfectly Normal Space is Completely Normal.
* Completely Normal implies $T_5$ by definition.
* Completely Normal Space is Normal.
* $T_5$ space is $T_4$.
* Normal implies $T_4... | Let $P_1$ and $P_2$ be [[Definition:Separation Axioms|separation axioms]] and let:
:$P_1 \implies P_2$
mean:
:If a [[Definition:Topological Space|topological space]] $T$ satsifies [[Definition:Separation Axioms|separation axiom]] $P_1$, then $T$ also satisfies [[Definition:Separation Axioms|separation axiom]] $P_2$.
... | The relevant justifications are listed as follows:
* [[Definition:Perfectly Normal Space|Perfectly Normal]] implies [[Definition:Perfectly T4 Space|Perfectly $T_4$]] by definition.
* [[Perfectly T4 Space is T5|Perfectly $T_4$ Space is $T_5$]]
* [[Perfectly Normal Space is Completely Normal]].
* [[Definition:Complet... | Sequence of Implications of Separation Axioms | https://proofwiki.org/wiki/Sequence_of_Implications_of_Separation_Axioms | https://proofwiki.org/wiki/Sequence_of_Implications_of_Separation_Axioms | [
"Separation Axioms"
] | [
"Definition:Tychonoff Separation Axioms",
"Definition:Topological Space",
"Definition:Tychonoff Separation Axioms",
"Definition:Tychonoff Separation Axioms",
"Definition:Tychonoff Separation Axioms",
"Definition:Perfectly Normal Space",
"Definition:Perfectly T4 Space",
"Definition:Completely Normal Sp... | [
"Definition:Perfectly Normal Space",
"Definition:Perfectly T4 Space",
"Perfectly T4 Space is T5",
"Perfectly Normal Space is Completely Normal",
"Definition:Completely Normal Space",
"Definition:T5 Space",
"Completely Normal Space is Normal",
"T5 Space is T4",
"Definition:Normal Space",
"Definitio... |
proofwiki-3475 | Sequence of Implications of Global Compactness Properties | Let $P_1$ and $P_2$ be topological compactness properties.
Let:
:$P_1 \implies P_2$
mean:
:If a topological space $T$ satisfies property $P_1$, then $T$ also satisfies property $P_2$.
Then the following sequence of implications holds:
{|
|-
| align="center" | ||
| align="center" | ||
| align="center" | Sequentially Com... | The relevant justifications are listed as follows:
* Sequentially Compact Space is Countably Compact
* Compact Space is Countably Compact
* Compact Space is $\sigma$-Compact
* $\sigma$-Compact Space is Lindelöf
* Countably Compact Space is Weakly Countably Compact
* Countably Compact Space is Pseudocompact
Then we have... | Let $P_1$ and $P_2$ be [[Definition:Topological Compactness|topological compactness properties]].
Let:
:$P_1 \implies P_2$
mean:
:If a [[Definition:Topological Space|topological space]] $T$ satisfies [[Definition:Property|property]] $P_1$, then $T$ also satisfies [[Definition:Property|property]] $P_2$.
Then the foll... | The relevant justifications are listed as follows:
* [[Sequentially Compact Space is Countably Compact]]
* [[Compact Space is Countably Compact]]
* [[Compact Space is Sigma-Compact Space|Compact Space is $\sigma$-Compact]]
* [[Sigma-Compact Space is Lindelöf Space|$\sigma$-Compact Space is Lindelöf]]
* [[Countably Com... | Sequence of Implications of Global Compactness Properties | https://proofwiki.org/wiki/Sequence_of_Implications_of_Global_Compactness_Properties | https://proofwiki.org/wiki/Sequence_of_Implications_of_Global_Compactness_Properties | [
"Sequence of Implications of Global Compactness Properties",
"Global Compactness Properties",
"Topological Compactness"
] | [
"Definition:Topological Compactness",
"Definition:Topological Space",
"Definition:Property",
"Definition:Property",
"Definition:Sequentially Compact Space",
"Definition:Compact Topological Space",
"Definition:Countably Compact Space",
"Definition:Pseudocompact Space",
"Definition:Sigma-Compact Space... | [
"Sequentially Compact Space is Countably Compact",
"Compact Space is Countably Compact",
"Compact Space is Sigma-Compact",
"Sigma-Compact Space is Lindelöf",
"Countably Compact Space is Weakly Countably Compact",
"Countably Compact Space is Pseudocompact",
"Compact Space is not necessarily Sequentially ... |
proofwiki-3476 | Sequence of Implications of Local Compactness Properties | Let $P_1$ and $P_2$ be topological compactness properties.
Let:
:$P_1 \implies P_2$
mean:
:If a topological space $T$ satisfies property $P_1$, then $T$ also satisfies property $P_2$.
Then the following sequence of implications holds:
{|
|-
| align="center" | Compact ||
| align="center" | $\implies$ ||
| align="center"... | The relevant justifications are listed as follows:
* Compact Space is Strongly Locally Compact.
* Strongly Locally Compact Space is Weakly Locally Compact.
* Locally Compact Space is Weakly Locally Compact.
* Compact Space is Weakly $\sigma$-Locally Compact.
* A weakly $\sigma$-locally compact is both weakly locally co... | Let $P_1$ and $P_2$ be [[Definition:Topological Compactness|topological compactness properties]].
Let:
:$P_1 \implies P_2$
mean:
:If a [[Definition:Topological Space|topological space]] $T$ satisfies [[Definition:Property|property]] $P_1$, then $T$ also satisfies [[Definition:Property|property]] $P_2$.
Then the foll... | The relevant justifications are listed as follows:
* [[Compact Space is Strongly Locally Compact]].
* [[Strongly Locally Compact Space is Weakly Locally Compact]].
* [[Locally Compact Space is Weakly Locally Compact]].
* [[Compact Space is Weakly Sigma-Locally Compact|Compact Space is Weakly $\sigma$-Locally Compact]]... | Sequence of Implications of Local Compactness Properties | https://proofwiki.org/wiki/Sequence_of_Implications_of_Local_Compactness_Properties | https://proofwiki.org/wiki/Sequence_of_Implications_of_Local_Compactness_Properties | [
"Sequence of Implications of Local Compactness Properties",
"Local Compactness Properties",
"Topological Compactness"
] | [
"Definition:Topological Compactness",
"Definition:Topological Space",
"Definition:Property",
"Definition:Property",
"Definition:Compact Topological Space",
"Definition:Strongly Locally Compact Space",
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:Weakly Locally Compact Space",
"Defini... | [
"Compact Space is Strongly Locally Compact",
"Strongly Locally Compact Space is Weakly Locally Compact",
"Locally Compact Space is Weakly Locally Compact",
"Compact Space is Weakly Sigma-Locally Compact",
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:Weakly Locally Compact Space",
"Defini... |
proofwiki-3477 | Equivalence of Definitions of Integral Dependence | Let $A$ be an extension of a commutative ring with unity $R$.
For $x \in A$, the following are equivalent:
{{begin-eqn}}
{{eqn | n = 1
| o =
| r = \) $x$ is integral over $R$ \(
}}
{{eqn | n = 2
| o =
| r = \) The $R$-module $R \sqbrk x$ is finitely generated \(
}}
{{eqn | n = 3
| o =
... | === $(1) \implies (2)$ ===
By hypothesis, there exist $r_0, \ldots, r_{n - 1} \in R$ such that:
:$x^n + r_{n - 1} x^{n - 1} + \cdots + r_1 x + r_0 = 0$
So the powers $x^k$, $k \ge n$ can be written as an $R$-linear combination of:
:$\set {1, \ldots, x^{n - 1} }$
Therefore this set generates $R \sqbrk x$.
{{qed|lemma}} | Let $A$ be an [[Definition:Ring Extension|extension]] of a [[Definition:Commutative and Unitary Ring|commutative ring with unity]] $R$.
For $x \in A$, the following are equivalent:
{{begin-eqn}}
{{eqn | n = 1
| o =
| r = \) $x$ is [[Definition:Integral Element of Ring Extension|integral]] over $R$ \(
}}
{... | === $(1) \implies (2)$ ===
By hypothesis, there exist $r_0, \ldots, r_{n - 1} \in R$ such that:
:$x^n + r_{n - 1} x^{n - 1} + \cdots + r_1 x + r_0 = 0$
So the powers $x^k$, $k \ge n$ can be written as an $R$-[[Definition:Linear Combination|linear combination]] of:
:$\set {1, \ldots, x^{n - 1} }$
Therefore this [[D... | Equivalence of Definitions of Integral Dependence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Integral_Dependence | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Integral_Dependence | [
"Algebraic Number Theory",
"Commutative Algebra"
] | [
"Definition:Ring Extension",
"Definition:Commutative and Unitary Ring",
"Definition:Integral Element of Ring Extension",
"Definition:Module over Ring",
"Definition:Finitely Generated Module",
"Definition:Subring",
"Definition:Finitely Generated Module",
"Definition:Subring",
"Definition:Finitely Gen... | [
"Definition:Linear Combination",
"Definition:Set",
"Definition:Generator of Ring"
] |
proofwiki-3478 | Transitivity of Finite Generation | Let $\struct { C, +_C, \circ_C}$ be a ring with unity, where $1_C$ denotes the unity of $C$.
Let $\struct { B, +_B, \circ_B}$ be a subring of $C$, such that $1_C \in B$.
Let $\struct { A, +_A, \circ_A}$ be a subring of $B$, such that $1_C \in A$.
Let $B' = \struct { B, +_B, \circ_B}_A$ be a finitely generated $A$-modul... | As $1_C \in B$, and $1_C \in A$, it follows that $A$ and $B$ are rings with unity.
Let $b_1, \ldots, b_n \in B$ generate $B'$ over $A$.
Let $c_1, \ldots, c_m \in C$ generate $C'$ over $B$.
Then for any $x \in C$, there exist $\beta_k \in B$, $k = 1, \ldots, m$ such that:
:$\ds x = \sum_{k \mathop = 1}^m \beta_k \circ_C... | Let $\struct { C, +_C, \circ_C}$ be a [[Definition:Ring with Unity|ring with unity]], where $1_C$ denotes the [[Definition:Unity of Ring|unity]] of $C$.
Let $\struct { B, +_B, \circ_B}$ be a [[Definition:Subring|subring]] of $C$, such that $1_C \in B$.
Let $\struct { A, +_A, \circ_A}$ be a [[Definition:Subring|subrin... | As $1_C \in B$, and $1_C \in A$, it follows that $A$ and $B$ are [[Definition:Ring with Unity|rings with unity]].
Let $b_1, \ldots, b_n \in B$ [[Definition:Generator of Module/Unitary|generate]] $B'$ over $A$.
Let $c_1, \ldots, c_m \in C$ [[Definition:Generator of Module/Unitary|generate]] $C'$ over $B$.
Then for a... | Transitivity of Finite Generation | https://proofwiki.org/wiki/Transitivity_of_Finite_Generation | https://proofwiki.org/wiki/Transitivity_of_Finite_Generation | [
"Commutative Algebra",
"Algebraic Number Theory"
] | [
"Definition:Ring with Unity",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Subring",
"Definition:Subring",
"Definition:Finitely Generated Module",
"Definition:Module over Ring",
"Definition:Scalar Multiplication",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Finitely Generate... | [
"Definition:Ring with Unity",
"Definition:Generator of Module/Unitary",
"Definition:Generator of Module/Unitary",
"Definition:Subring",
"Definition:Ring (Abstract Algebra)/Product",
"Definition:Linear Combination",
"Definition:Scalar/Module",
"Definition:Generator of Module",
"Category:Commutative A... |
proofwiki-3479 | Integral Closure is Subring | Let $A$ be an extension of a commutative ring with unity $\struct {R, +, \circ}$.
Let $C$ be the integral closure of $R$ in $A$.
Then $C$ is a subring of $A$. | For each $r \in R$, we have $r$ is a zero of $a - r = 0$.
{{MissingLinks|zero}}
Hence $R \subseteq C$, and in particular, $C$ is non-empty.
By Subring Test it is sufficient to prove that if $x, y \in C$ then $x + \paren {-y}, x \circ y \in C$.
Let $x, y \in C$.
By Equivalent Definitions of Integral Dependence: $(1) \im... | Let $A$ be an [[Definition:Ring Extension|extension]] of a [[Definition:Commutative and Unitary Ring|commutative ring with unity]] $\struct {R, +, \circ}$.
Let $C$ be the [[Definition:Integral Closure|integral closure]] of $R$ in $A$.
Then $C$ is a [[Definition:Subring|subring]] of $A$. | For each $r \in R$, we have $r$ is a zero of $a - r = 0$.
{{MissingLinks|zero}}
Hence $R \subseteq C$, and in particular, $C$ is [[Definition:Non-Empty Set|non-empty]].
By [[Subring Test]] it is sufficient to prove that if $x, y \in C$ then $x + \paren {-y}, x \circ y \in C$.
Let $x, y \in C$.
By [[Equivalent Defin... | Integral Closure is Subring | https://proofwiki.org/wiki/Integral_Closure_is_Subring | https://proofwiki.org/wiki/Integral_Closure_is_Subring | [
"Commutative Algebra",
"Algebraic Number Theory"
] | [
"Definition:Ring Extension",
"Definition:Commutative and Unitary Ring",
"Definition:Integral Closure",
"Definition:Subring"
] | [
"Definition:Non-Empty Set",
"Subring Test",
"Equivalence of Definitions of Integral Dependence",
"Definition:Finitely Generated Module",
"Definition:Finitely Generated Module",
"Definition:Finitely Generated Module",
"Definition:Ring Extension",
"Transitivity of Finite Generation",
"Definition:Finit... |
proofwiki-3480 | Localization of Ring is Unique | Let $A$ be a commutative ring with unity.
Let $S \subseteq A$ be a multiplicatively closed subset.
Let $\struct {A_S, \iota}$ and $\struct {\tilde A_S, \tilde \iota}$ both satisfy the definition of the localization of $A$ at $S$.
Then there is a canonical isomorphism $\phi: A_S \to \tilde A_S$. | By the definition of localization, there exist unique homomorphisms:
:$g : A_S \to \tilde A_S$
:$h : \tilde A_S \to A_S$
such that:
:$h \circ \iota = \tilde \iota$
and:
:$g \circ \tilde \iota = \iota$
Therefore:
:$\tilde \iota = h \circ \iota = h \circ \paren {g \circ \tilde \iota} = \paren {h \circ g} \circ \tilde \io... | Let $A$ be a [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Let $S \subseteq A$ be a [[Definition:Multiplicatively Closed Subset of Ring|multiplicatively closed subset]].
Let $\struct {A_S, \iota}$ and $\struct {\tilde A_S, \tilde \iota}$ both satisfy the definition of the [[Definition:Local... | By the definition of [[Definition:Localization of Ring|localization]], there exist unique [[Definition:Ring Homomorphism|homomorphisms]]:
:$g : A_S \to \tilde A_S$
:$h : \tilde A_S \to A_S$
such that:
:$h \circ \iota = \tilde \iota$
and:
:$g \circ \tilde \iota = \iota$
Therefore:
:$\tilde \iota = h \circ \iota = h \ci... | Localization of Ring is Unique | https://proofwiki.org/wiki/Localization_of_Ring_is_Unique | https://proofwiki.org/wiki/Localization_of_Ring_is_Unique | [
"Localization of Rings"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Multiplicatively Closed Subset of Ring",
"Definition:Localization of Ring",
"Definition:Canonical Isomorphism"
] | [
"Definition:Localization of Ring",
"Definition:Ring Homomorphism",
"Definition:Identity Mapping",
"Definition:Identity Mapping",
"Definition:Identity Mapping",
"Bijection iff Left and Right Inverse",
"Definition:Bijection",
"Definition:Ring Homomorphism",
"Definition:Isomorphism (Abstract Algebra)/R... |
proofwiki-3481 | Spectrum of Ring is Nonempty | Let $A$ be a non-trivial commutative ring with unity.
Then its prime spectrum is non-empty:
:$\Spec A \ne \O$ | This is a reformulation of Ring with Unity has Prime Ideal.
{{qed}} | Let $A$ be a [[Definition:Non-Trivial Ring|non-trivial]] [[Definition:Commutative and Unitary Ring|commutative ring with unity]].
Then its [[Definition:Spectrum of Ring|prime spectrum]] is [[Definition:Non-Empty Set|non-empty]]:
:$\Spec A \ne \O$ | This is a reformulation of [[Ring with Unity has Prime Ideal]].
{{qed}} | Spectrum of Ring is Nonempty | https://proofwiki.org/wiki/Spectrum_of_Ring_is_Nonempty | https://proofwiki.org/wiki/Spectrum_of_Ring_is_Nonempty | [
"Commutative Algebra"
] | [
"Definition:Non-Trivial Ring",
"Definition:Commutative and Unitary Ring",
"Definition:Prime Spectrum of Ring",
"Definition:Non-Empty Set"
] | [
"Ring with Unity has Prime Ideal"
] |
proofwiki-3482 | Compact Space is Strongly Locally Compact | Let $T = \struct {S, \tau}$ be a compact space.
Then $T$ is a strongly locally compact space. | {{Recall|Strongly Locally Compact Space|strongly locally compact space}}
{{:Definition:Strongly Locally Compact Space/Definition 1}}
Let $T = \struct {S, \tau}$ be compact.
From Underlying Set of Topological Space is Clopen, $S$ is clopen in $T$.
From Closed Set Equals its Closure, $S = S^-$.
So every point of $S$ is c... | Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact space]].
Then $T$ is a [[Definition:Strongly Locally Compact Space|strongly locally compact space]]. | {{Recall|Strongly Locally Compact Space|strongly locally compact space}}
{{:Definition:Strongly Locally Compact Space/Definition 1}}
Let $T = \struct {S, \tau}$ be [[Definition:Compact Topological Space|compact]].
From [[Underlying Set of Topological Space is Clopen]], $S$ is [[Definition:Clopen Set|clopen]] in $T$.
... | Compact Space is Strongly Locally Compact | https://proofwiki.org/wiki/Compact_Space_is_Strongly_Locally_Compact | https://proofwiki.org/wiki/Compact_Space_is_Strongly_Locally_Compact | [
"Compact Topological Spaces",
"Strongly Locally Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Compact Topological Space",
"Definition:Strongly Locally Compact Space"
] | [
"Definition:Compact Topological Space",
"Underlying Set of Topological Space is Clopen",
"Definition:Clopen Set",
"Set is Closed iff Equals Topological Closure",
"Definition:Open Set/Topology",
"Definition:Closure (Topology)",
"Definition:Compact Topological Space/Subspace",
"Definition:Compact Topolo... |
proofwiki-3483 | Compact Space is Weakly Sigma-Locally Compact | Let $T = \struct {S, \tau}$ be a compact space.
Then $T$ is a weakly $\sigma$-locally compact space. | {{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
{{:Definition:Weakly Sigma-Locally Compact Space}}
Let $T = \struct {S, \tau}$ be a compact space.
We have:
{{begin-itemize}}
{{item||Compact Space is $\sigma$-Compact}}
{{item||Compact Space is Weakly Locally Compact}}
{{end-itemize}}
... | Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact space]].
Then $T$ is a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space]]. | {{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
{{:Definition:Weakly Sigma-Locally Compact Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Compact Topological Space|compact space]].
We have:
{{begin-itemize}}
{{item||[[Compact Space is Sigma-Compact|Compact Space is $\sigma$-... | Compact Space is Weakly Sigma-Locally Compact | https://proofwiki.org/wiki/Compact_Space_is_Weakly_Sigma-Locally_Compact | https://proofwiki.org/wiki/Compact_Space_is_Weakly_Sigma-Locally_Compact | [
"Weakly Sigma-Locally Compact Spaces",
"Compact Topological Spaces",
"Sequence of Implications of Local Compactness Properties",
"Sequence of Implications of Metric Space Compactness Properties"
] | [
"Definition:Compact Topological Space",
"Definition:Weakly Sigma-Locally Compact Space"
] | [
"Definition:Compact Topological Space",
"Compact Space is Sigma-Compact",
"Compact Space is Weakly Locally Compact",
"Definition:Weakly Sigma-Locally Compact Space"
] |
proofwiki-3484 | Sequence of Implications of Paracompactness Properties | Let $P_1$ and $P_2$ be paracompactness properties.
Let:
:$P_1 \implies P_2$
mean:
:If a topological space $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$.
Then the following sequence of implications holds:
{|
|-
| align="center" | Fully Normal ||
| align="center" | ||
| align="center" | Compact ||
... | The relevant justifications are listed as follows:
* Compact Space is Countably Compact
* Compact Space is Paracompact
* Fully $T_4$ Space is Paracompact
* Countably Compact Space is Countably Paracompact
* Paracompact Space is Countably Paracompact
* Fully Normal Space is Fully $T_4$ by definition
* Fully $T_4$ Space ... | Let $P_1$ and $P_2$ be [[Definition:Paracompactness Property|paracompactness properties]].
Let:
:$P_1 \implies P_2$
mean:
:If a [[Definition:Topological Space|topological space]] $T$ satsifies property $P_1$, then $T$ also satisfies property $P_2$.
Then the following sequence of implications holds:
{|
|-
| align="... | The relevant justifications are listed as follows:
* [[Compact Space is Countably Compact]]
* [[Compact Space is Paracompact]]
* [[Fully T4 Space is Paracompact|Fully $T_4$ Space is Paracompact]]
* [[Countably Compact Space is Countably Paracompact]]
* [[Paracompact Space is Countably Paracompact]]
* [[Definition:Full... | Sequence of Implications of Paracompactness Properties | https://proofwiki.org/wiki/Sequence_of_Implications_of_Paracompactness_Properties | https://proofwiki.org/wiki/Sequence_of_Implications_of_Paracompactness_Properties | [
"Sequence of Implications of Paracompactness Properties",
"Paracompactness Properties",
"Topological Compactness"
] | [
"Definition:Paracompactness Property",
"Definition:Topological Space",
"Definition:Fully Normal Space",
"Definition:Compact Topological Space",
"Definition:Countably Compact Space",
"Definition:Fully T4 Space",
"Definition:Paracompact Space",
"Definition:Countably Paracompact Space",
"Definition:T4 ... | [
"Compact Space is Countably Compact",
"Compact Space is Paracompact",
"Fully T4 Space is Paracompact",
"Countably Compact Space is Countably Paracompact",
"Paracompact Space is Countably Paracompact",
"Definition:Fully Normal Space",
"Definition:Fully T4 Space",
"Fully T4 Space is T4",
"Paracompact ... |
proofwiki-3485 | Sequence of Implications of Compactness Properties in Hausdorff Space | Let $P_1$ and $P_2$ be compactness properties.
Let:
:$P_1 \implies P_2$
mean:
:If a $T_2$ (Hausdorff) space $T$ satisfies property $P_1$, then $T$ also satisfies property $P_2$.
Then the following sequence of implications holds:
{|
|-
| align="center" | Fully $T_4$ ||
| align="center" | $\iff$ ||
| align="center" | Par... | From $T_2$ Space is $T_1$ and $T_1$ Space is $T_0$, we note that in a $T_2$ (Hausdorff) space:
:A fully $T_4$ space is a fully normal space
:A $T_4$ space is a normal space
:A $T_{3 \frac 1 2}$ space is a completely regular space
:A $T_3$ space is a regular space
all by definition.
From $T_2$ Space is Fully $T_4$ iff P... | Let $P_1$ and $P_2$ be [[Definition:Compact Topological Space|compactness properties]].
Let:
:$P_1 \implies P_2$
mean:
:If a [[Definition:Hausdorff Space|$T_2$ (Hausdorff) space]] $T$ satisfies property $P_1$, then $T$ also satisfies property $P_2$.
Then the following sequence of implications holds:
{|
|-
| align=... | From [[T2 Space is T1|$T_2$ Space is $T_1$]] and [[T1 Space is T0|$T_1$ Space is $T_0$]], we note that in a [[Definition:Hausdorff Space|$T_2$ (Hausdorff) space]]:
:A [[Definition:Fully T4 Space|fully $T_4$ space]] is a [[Definition:Fully Normal Space|fully normal space]]
:A [[Definition:T4 Space|$T_4$ space]] is a [... | Sequence of Implications of Compactness Properties in Hausdorff Space | https://proofwiki.org/wiki/Sequence_of_Implications_of_Compactness_Properties_in_Hausdorff_Space | https://proofwiki.org/wiki/Sequence_of_Implications_of_Compactness_Properties_in_Hausdorff_Space | [
"Sequence of Implications of Compactness Properties in Hausdorff Space",
"Hausdorff Spaces",
"Topological Compactness"
] | [
"Definition:Compact Topological Space",
"Definition:T2 Space",
"Definition:Fully T4 Space",
"Definition:Paracompact Space",
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:T4 Space",
"Definition:Weakly Locally Compact Space",
"Definition:T3.5 Space",
"Definition:First-Countable Space",
... | [
"T2 Space is T1",
"T1 Space is T0",
"Definition:T2 Space",
"Definition:Fully T4 Space",
"Definition:Fully Normal Space",
"Definition:T4 Space",
"Definition:Normal Space",
"Definition:T3.5 Space",
"Definition:Completely Regular Space",
"Definition:T3 Space",
"Definition:Regular Space",
"T2 Spac... |
proofwiki-3486 | Zero Locus of Set is Zero Locus of Generated Ideal | Let $k$ be a field.
Let $n \ge 1$ be a natural number.
Let $A = k \sqbrk {X_1, \ldots, X_n}$ be the ring of polynomial functions in $n$ variables over $k$.
Let $T \subseteq A$ be a subset, and $\map V T$ the zero locus of $T$.
Let $J = \ideal T$ be the ideal generated by $T$.
Then:
:$\map V T = \map V J$ | Let $x \in \map V T$, so $\map f x = 0$ for all $f \in T$.
By definition, $J$ is the set of linear combinations of elements of $T$ over $k$.
So any $g \in J$ is of the form:
:$g = k_1 t_1 + \cdots + k_r t_r$
with $k_i \in k$ and $t_i \in T$.
Therefore:
{{begin-eqn}}
{{eqn | l = \map g x
| r = k_1 \map {t_1} x + \... | Let $k$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $n \ge 1$ be a [[Definition:Natural Number|natural number]].
Let $A = k \sqbrk {X_1, \ldots, X_n}$ be the [[Definition:Ring of Polynomial Functions|ring of polynomial functions]] in $n$ variables over $k$.
Let $T \subseteq A$ be a [[Definition:Subset|su... | Let $x \in \map V T$, so $\map f x = 0$ for all $f \in T$.
By definition, $J$ is the set of [[Definition:Linear Combination|linear combinations]] of elements of $T$ over $k$.
So any $g \in J$ is of the form:
:$g = k_1 t_1 + \cdots + k_r t_r$
with $k_i \in k$ and $t_i \in T$.
Therefore:
{{begin-eqn}}
{{eqn | l = \... | Zero Locus of Set is Zero Locus of Generated Ideal | https://proofwiki.org/wiki/Zero_Locus_of_Set_is_Zero_Locus_of_Generated_Ideal | https://proofwiki.org/wiki/Zero_Locus_of_Set_is_Zero_Locus_of_Generated_Ideal | [
"Algebraic Geometry"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Natural Numbers",
"Definition:Ring of Polynomial Functions",
"Definition:Subset",
"Definition:Zero Locus of Set of Polynomials",
"Definition:Ideal of Ring",
"Definition:Generator of Ideal of Ring"
] | [
"Definition:Linear Combination",
"Category:Algebraic Geometry"
] |
proofwiki-3487 | Sequence of Implications of Compactness Properties in T3 Space | Let $P_1$ and $P_2$ be compactness properties.
Let:
:$P_1 \implies P_2$
mean:
:If a $T_3$ space $T$ satisfies property $P_1$, then $T$ also satisfies property $P_2$.
Then the following sequence of implications holds:
{|
|-
| align="center" | Second-Countable ||
| align="center" | $\implies$ ||
| align="center" | Lindel... | The justifications are listed as follows:
:Second-Countable $T_3$ Space is $T_5$
:Second-Countable Space is Lindelöf
:$T_3$ Lindelöf Space is Paracompact
:$T_3$ Space is Fully $T_4$ iff Paracompact
:Fully $T_4$ Space is $T_4$
:$T_5$ Space is $T_4$
{{qed}} | Let $P_1$ and $P_2$ be [[Definition:Compact Topological Space|compactness properties]].
Let:
:$P_1 \implies P_2$
mean:
:If a [[Definition:T3 Space|$T_3$ space]] $T$ satisfies property $P_1$, then $T$ also satisfies property $P_2$.
Then the following sequence of implications holds:
{|
|-
| align="center" | [[Defini... | The justifications are listed as follows:
:[[Second-Countable T3 Space is T5|Second-Countable $T_3$ Space is $T_5$]]
:[[Second-Countable Space is Lindelöf]]
:[[T3 Lindelöf Space is Paracompact|$T_3$ Lindelöf Space is Paracompact]]
:[[T3 Space is Fully T4 iff Paracompact|$T_3$ Space is Fully $T_4$ iff Paracompact]]
... | Sequence of Implications of Compactness Properties in T3 Space | https://proofwiki.org/wiki/Sequence_of_Implications_of_Compactness_Properties_in_T3_Space | https://proofwiki.org/wiki/Sequence_of_Implications_of_Compactness_Properties_in_T3_Space | [
"Sequence of Implications of Compactness Properties in T3 Space",
"T3 Spaces",
"Topological Compactness"
] | [
"Definition:Compact Topological Space",
"Definition:T3 Space",
"Definition:Second-Countable Space",
"Definition:Lindelöf Space",
"Definition:Fully T4 Space",
"Definition:Paracompact Space",
"Definition:T5 Space",
"Definition:T4 Space"
] | [
"Second-Countable T3 Space is T5",
"Second-Countable Space is Lindelöf",
"T3 Lindelöf Space is Paracompact",
"T3 Space is Fully T4 iff Paracompact",
"Fully T4 Space is T4",
"T5 Space is T4"
] |
proofwiki-3488 | Finite Product of Sigma-Compact Spaces is Sigma-Compact | Let $n \in \Z_{>0}$ be a (strictly) positive integer.
Let $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$ be a finite set of topological spaces.
Let $\ds \struct {S, \tau} = \prod_{i \mathop = 1}^n \struct {S_i, \tau_i}$ be the product space of $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$.
Let each of $\struct {S_i, \ta... | Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be $\sigma$-compact.
Then $\ds S_1 = \bigcup_{n \mathop = 1}^\infty C_n$ and $\ds S_2 = \bigcup_{n \mathop = 1}^\infty D_n$ where all the $C_n$ and $D_n$ are compact sets.
From Tychonoff's Theorem, the set $\KK = \set {C_n \times D_m: n, m \in \N}$ is a countable ... | Let $n \in \Z_{>0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$ be a [[Definition:Finite Set|finite set]] of [[Definition:Topological Space|topological spaces]].
Let $\ds \struct {S, \tau} = \prod_{i \mathop = 1}^n \struct {S_i, \tau_i}$... | Let $\struct {S_1, \tau_1}$ and $\struct {S_2, \tau_2}$ be [[Definition:Sigma-Compact Space|$\sigma$-compact]].
Then $\ds S_1 = \bigcup_{n \mathop = 1}^\infty C_n$ and $\ds S_2 = \bigcup_{n \mathop = 1}^\infty D_n$ where all the $C_n$ and $D_n$ are [[Definition:Compact Topological Subspace|compact sets]].
From [[Tych... | Finite Product of Sigma-Compact Spaces is Sigma-Compact | https://proofwiki.org/wiki/Finite_Product_of_Sigma-Compact_Spaces_is_Sigma-Compact | https://proofwiki.org/wiki/Finite_Product_of_Sigma-Compact_Spaces_is_Sigma-Compact | [
"Sigma-Compact Spaces",
"Product Spaces"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Finite Set",
"Definition:Topological Space",
"Definition:Product Space (Topology)",
"Definition:Sigma-Compact Space",
"Definition:Sigma-Compact Space"
] | [
"Definition:Sigma-Compact Space",
"Definition:Compact Topological Space/Subspace",
"Tychonoff's Theorem",
"Definition:Countable Set",
"Definition:Compact Topological Space/Subspace",
"Definition:Product Topology",
"Definition:Sigma-Compact Space",
"Definition:Sigma-Compact Space"
] |
proofwiki-3489 | Countable Product of Sequentially Compact Spaces is Sequentially Compact | Let $I$ be an indexing set with countable cardinality.
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.
Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha,... | {{Tidy|By the very nature of the proof, the indices are a pain to look at.}}
Let $\sequence { {\mathbf x}_n}$ be a sequence in $S$.
That is, for each $n \in \N$, ${\mathbf x}_n$ is a tuple $\family {x_n^i}_{i \mathop \in I}$ indexed by $I$.
We claim there is a strictly increasing sequence $\sequence {n_r}$ in $\N$ such... | Let $I$ be an [[Definition:Indexing Set|indexing set]] with [[Definition:Countable Set|countable cardinality]].
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] [[Definition:Indexed Family|indexed]] ... | {{Tidy|By the very nature of the proof, the indices are a pain to look at.}}
Let $\sequence { {\mathbf x}_n}$ be a [[Definition:Sequence|sequence]] in $S$.
That is, for each $n \in \N$, ${\mathbf x}_n$ is a tuple $\family {x_n^i}_{i \mathop \in I}$ indexed by $I$.
We claim there is a [[Definition:Strictly Increasing... | Countable Product of Sequentially Compact Spaces is Sequentially Compact | https://proofwiki.org/wiki/Countable_Product_of_Sequentially_Compact_Spaces_is_Sequentially_Compact | https://proofwiki.org/wiki/Countable_Product_of_Sequentially_Compact_Spaces_is_Sequentially_Compact | [
"Sequentially Compact Spaces",
"Product Spaces"
] | [
"Definition:Indexing Set",
"Definition:Countable Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Sequentially Compact Space",
"Definition:Sequentially Compact Space"
] | [
"Definition:Sequence",
"Definition:Strictly Increasing/Sequence",
"Definition:Subsequence",
"Sequence on Product Space Converges to Point iff Projections Converge to Projections of Point",
"Definition:Strictly Increasing/Sequence",
"Definition:Subsequence",
"Definition:Enumeration",
"Definition:Strict... |
proofwiki-3490 | Finite Product of Weakly Locally Compact Spaces is Weakly Locally Compact | Let $n \in \Z_{\ge 0}$ be a (strictly) positive integer.
Let $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$ be a finite set of topological spaces.
Let $\ds \struct {S, \tau} = \prod_{i \mathop = 1}^n \struct {S_i, \tau_i}$ be the product space of $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$.
Let each of $\struct {S_i, ... | Pick any $\tuple {x_1, \dots, x_n} \in S$.
Since each of $\struct {S_i, \tau_i}$ is weakly locally compact, $x_i$ has a compact neighborhood.
That is, there is a compact set $N_i$ such that:
:$\exists U_i \in \tau_i: x_i \in U_i \subseteq N_i$
From Cartesian Product of Family of Subsets:
:$\ds \tuple {x_1, \dots, x_n} ... | Let $n \in \Z_{\ge 0}$ be a [[Definition:Strictly Positive Integer|(strictly) positive integer]].
Let $\set {\struct {S_i, \tau_i}: 1 \le i \le n}$ be a [[Definition:Finite Set|finite set]] of [[Definition:Topological Space|topological spaces]].
Let $\ds \struct {S, \tau} = \prod_{i \mathop = 1}^n \struct {S_i, \tau_... | Pick any $\tuple {x_1, \dots, x_n} \in S$.
Since each of $\struct {S_i, \tau_i}$ is [[Definition:Weakly Locally Compact Space|weakly locally compact]], $x_i$ has a [[Definition:Compact Topological Space|compact]] [[Definition:Neighborhood of Point|neighborhood]].
That is, there is a [[Definition:Compact Topological S... | Finite Product of Weakly Locally Compact Spaces is Weakly Locally Compact | https://proofwiki.org/wiki/Finite_Product_of_Weakly_Locally_Compact_Spaces_is_Weakly_Locally_Compact | https://proofwiki.org/wiki/Finite_Product_of_Weakly_Locally_Compact_Spaces_is_Weakly_Locally_Compact | [
"Weakly Locally Compact Spaces",
"Product Spaces"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Finite Set",
"Definition:Topological Space",
"Definition:Product Space (Topology)",
"Definition:Weakly Locally Compact Space",
"Definition:Weakly Locally Compact Space"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:Compact Topological Space",
"Definition:Neighborhood (Topology)/Point",
"Definition:Compact Topological Space",
"Cartesian Product of Subsets/Family of Subsets",
"Definition:Product Space (Topology)",
"Topological Product of Compact Spaces/Finite Pr... |
proofwiki-3491 | Countable Product of First-Countable Spaces is First-Countable | Let $I$ be an indexing set with countable cardinality.
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.
Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha,... | Let $x = \family {x_\alpha} \in S$.
Let $\BB_\alpha = \set {N^\alpha_i : i \in \N}$ be a countable local basis for $x_\alpha$ in $\struct {S_\alpha, \tau_\alpha}$ for each $\alpha \in I$.
Let $\pr_\alpha$ denote the projection of $S$ onto $S_\alpha$.
Let $\LL_\alpha = \set {\map {\pr_\alpha^{-1} } {N^\alpha_i}: N^\alph... | Let $I$ be an [[Definition:Indexing Set|indexing set]] with [[Definition:Countable Set|countable cardinality]].
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] [[Definition:Indexed Family|indexed]] ... | Let $x = \family {x_\alpha} \in S$.
Let $\BB_\alpha = \set {N^\alpha_i : i \in \N}$ be a [[Definition:Countable Set|countable]] [[Definition:Local Basis|local basis]] for $x_\alpha$ in $\struct {S_\alpha, \tau_\alpha}$ for each $\alpha \in I$.
Let $\pr_\alpha$ denote the [[Definition:Projection|projection]] of $S$ on... | Countable Product of First-Countable Spaces is First-Countable | https://proofwiki.org/wiki/Countable_Product_of_First-Countable_Spaces_is_First-Countable | https://proofwiki.org/wiki/Countable_Product_of_First-Countable_Spaces_is_First-Countable | [
"First-Countable Spaces",
"Product Spaces",
"Countable Sets"
] | [
"Definition:Indexing Set",
"Definition:Countable Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:First-Countable Space",
"Definition:First-Countable Space"
] | [
"Definition:Countable Set",
"Definition:Local Basis",
"Definition:Projection",
"Product Space Local Basis Induced from Factor Spaces Local Bases",
"Definition:Local Basis",
"Definition:Product Space (Topology)",
"Definition:Countable Set",
"Definition:Finite Set",
"Definition:Cartesian Product",
"... |
proofwiki-3492 | Countable Product of Second-Countable Spaces is Second-Countable | Let $I$ be an indexing set with countable cardinality.
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.
Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha,... | Let $\BB_\alpha = \set {N^\alpha_i : i \in \N}$ be a countable basis for $\tau_\alpha$ for each $\alpha \in I$.
Let $\pr_\alpha$ denote the projection of $S$ onto $S_\alpha$.
Let $\LL_\alpha = \set {\map {\pr_\alpha^{-1} } {N^\alpha_i}: N^\alpha_i \in B_\alpha}$.
Let $\ds \KK_J = \set {\bigcap_{\alpha \mathop \in J} L_... | Let $I$ be an [[Definition:Indexing Set|indexing set]] with [[Definition:Countable Set|countable cardinality]].
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] [[Definition:Indexed Family|indexed]] ... | Let $\BB_\alpha = \set {N^\alpha_i : i \in \N}$ be a [[Definition:Countable Basis|countable basis]] for $\tau_\alpha$ for each $\alpha \in I$.
Let $\pr_\alpha$ denote the [[Definition:Projection|projection]] of $S$ onto $S_\alpha$.
Let $\LL_\alpha = \set {\map {\pr_\alpha^{-1} } {N^\alpha_i}: N^\alpha_i \in B_\alpha... | Countable Product of Second-Countable Spaces is Second-Countable | https://proofwiki.org/wiki/Countable_Product_of_Second-Countable_Spaces_is_Second-Countable | https://proofwiki.org/wiki/Countable_Product_of_Second-Countable_Spaces_is_Second-Countable | [
"Second-Countable Spaces",
"Product Spaces",
"Countable Sets"
] | [
"Definition:Indexing Set",
"Definition:Countable Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Second-Countable Space",
"Definition:Second-Countable Space"
] | [
"Definition:Countable Basis",
"Definition:Projection",
"Product Space Basis Induced from Factor Space Bases",
"Definition:Basis (Topology)",
"Definition:Product Space (Topology)",
"Definition:Countable Set",
"Definition:Finite Set",
"Definition:Cartesian Product",
"Definition:Countable Set",
"Defi... |
proofwiki-3493 | Countable Product of Separable Spaces is Separable | Let $I$ be an indexing set with countable cardinality.
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.
Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha,... | For each $\alpha \in I$, let $D_\alpha$ be a countable everywhere dense subset of $\struct {S_\alpha, \tau_\alpha}$.
Let $D = \ds \prod_{\alpha \mathop \in I} D_\alpha$.
From Countable Union of Countable Sets is Countable, $D$ is a countable set of $S$.
From Natural Basis of Product Topology, the set $\BB$ of cartesian... | Let $I$ be an [[Definition:Indexing Set|indexing set]] with [[Definition:Countable Set|countable cardinality]].
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] [[Definition:Indexed Family|indexed]] ... | For each $\alpha \in I$, let $D_\alpha$ be a [[Definition:Countable Set|countable]] [[Definition:Everywhere Dense|everywhere dense subset]] of $\struct {S_\alpha, \tau_\alpha}$.
Let $D = \ds \prod_{\alpha \mathop \in I} D_\alpha$.
From [[Countable Union of Countable Sets is Countable]], $D$ is a [[Definition:Countabl... | Countable Product of Separable Spaces is Separable | https://proofwiki.org/wiki/Countable_Product_of_Separable_Spaces_is_Separable | https://proofwiki.org/wiki/Countable_Product_of_Separable_Spaces_is_Separable | [
"Separable Spaces",
"Product Topology",
"Countable Sets"
] | [
"Definition:Indexing Set",
"Definition:Countable Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Separable Space",
"Definition:Separable Space"
] | [
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Countable Union of Countable Sets is Countable",
"Definition:Countable Set",
"Natural Basis of Product Topology",
"Definition:Set",
"Definition:Cartesian Product",
"Definition:Index",
"Definition:Basis (Topology)/Analytic Basis",
"Open Se... |
proofwiki-3494 | Compactness Properties Preserved under Continuous Surjection | Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces.
Let $\phi: T_A \to T_B$ be a continuous surjection.
If $T_A$ has one of the following properties, then $T_B$ has the same property:
:Compact Space
:$\sigma$-Compact Space
:Countable Compact Space
:Sequential Compact Space
:Lindel... | === Proof for Compactness ===
{{:Compactness is Preserved under Continuous Surjection}} | Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be [[Definition:Topological Space|topological spaces]].
Let $\phi: T_A \to T_B$ be a [[Definition:Everywhere Continuous Mapping (Topology)|continuous]] [[Definition:Surjection|surjection]].
If $T_A$ has one of the following properties, then $T_B$ ha... | === [[Compactness is Preserved under Continuous Surjection|Proof for Compactness]] ===
{{:Compactness is Preserved under Continuous Surjection}} | Compactness Properties Preserved under Continuous Surjection | https://proofwiki.org/wiki/Compactness_Properties_Preserved_under_Continuous_Surjection | https://proofwiki.org/wiki/Compactness_Properties_Preserved_under_Continuous_Surjection | [
"Compactness Properties Preserved under Continuous Surjection",
"Compact Topological Spaces",
"Continuous Mappings",
"Surjections"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Surjection",
"Definition:Compact Topological Space",
"Definition:Sigma-Compact Space",
"Definition:Countably Compact Space",
"Definition:Sequentially Compact Space",
"Definition:Lindelöf Space"
] | [
"Compactness is Preserved under Continuous Surjection"
] |
proofwiki-3495 | Projection from Product Topology is Open | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the product topology on $S$.
Let $\pr_1: T \to T_1$ and $\pr_2: T \to T_2$ be the first and second projections from $T$ onto its fa... | Let $U \in \tau$.
It follows from the definition of product topology that $U$ can be expressed as:
:$\ds U = \bigcup_{j \mathop \in J} \bigcap_{k \mathop = 1}^{n_j} \map {\pr_{i_{k, j} }^{-1} } {U_{k, j} }$
where:
:$J$ is an arbitrary index set
:$n_j \in \N$
:$i_{k, j} \in I$
:$U_{k, j} \in \tau_{i_{k, j} }$.
For all $... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $T = \struct {T_1 \times T_2, \tau}$ be the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] of $T_1$ and $T_2$, where $\tau$ is the [[Definition:Product Topology on... | Let $U \in \tau$.
It follows from the definition of [[Definition:Product Topology|product topology]] that $U$ can be expressed as:
:$\ds U = \bigcup_{j \mathop \in J} \bigcap_{k \mathop = 1}^{n_j} \map {\pr_{i_{k, j} }^{-1} } {U_{k, j} }$
where:
:$J$ is an arbitrary [[Definition:Indexing Set|index set]]
:$n_j \in \N... | Projection from Product Topology is Open/General Result/Proof | https://proofwiki.org/wiki/Projection_from_Product_Topology_is_Open | https://proofwiki.org/wiki/Projection_from_Product_Topology_is_Open/General_Result/Proof | [
"Projection from Product Topology is Open",
"Projection from Product Topology is Open and Continuous",
"Product Topology",
"Open Mappings",
"Projections"
] | [
"Definition:Topological Space",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Product Topology/Two Factor Spaces",
"Definition:Projection (Mapping Theory)",
"Definition:Product Topology/Factor Space",
"Definition:Open Mapping"
] | [
"Definition:Product Topology",
"Definition:Indexing Set",
"Image of Union under Relation/Family of Sets",
"Cartesian Product of Intersections/General Case",
"Definition:Open Mapping"
] |
proofwiki-3496 | Projection from Product Topology is Open | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $T = \struct {T_1 \times T_2, \tau}$ be the product space of $T_1$ and $T_2$, where $\tau$ is the product topology on $S$.
Let $\pr_1: T \to T_1$ and $\pr_2: T \to T_2$ be the first and second projections from $T$ onto its fa... | Let $U \in \tau$.
It follows from the definition of product topology that $U$ can be expressed as:
:$\ds U = \bigcup_{j \mathop \in J} \bigcap_{k \mathop = 1}^{n_j} \map {\pr_{i_{k,j} }^{-1}} { U_{k,j} }$
where $J$ is an arbitrary index set, $n_j \in \N$, $i_{k, j} \in \set {1, 2}$, and $U_{k, j} \in \tau_{i_{k, j} }$.... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $T = \struct {T_1 \times T_2, \tau}$ be the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] of $T_1$ and $T_2$, where $\tau$ is the [[Definition:Product Topology on... | Let $U \in \tau$.
It follows from the definition of [[Definition:Product Topology on Two Factor Spaces|product topology]] that $U$ can be expressed as:
:$\ds U = \bigcup_{j \mathop \in J} \bigcap_{k \mathop = 1}^{n_j} \map {\pr_{i_{k,j} }^{-1}} { U_{k,j} }$
where $J$ is an arbitrary [[Definition:Indexing Set|index s... | Projection from Product Topology is Open/Proof 1 | https://proofwiki.org/wiki/Projection_from_Product_Topology_is_Open | https://proofwiki.org/wiki/Projection_from_Product_Topology_is_Open/Proof_1 | [
"Projection from Product Topology is Open",
"Projection from Product Topology is Open and Continuous",
"Product Topology",
"Open Mappings",
"Projections"
] | [
"Definition:Topological Space",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Product Topology/Two Factor Spaces",
"Definition:Projection (Mapping Theory)",
"Definition:Product Topology/Factor Space",
"Definition:Open Mapping"
] | [
"Definition:Product Topology/Two Factor Spaces",
"Definition:Indexing Set",
"Definition:Projection (Mapping Theory)",
"Image of Union under Relation/Family of Sets",
"Cartesian Product of Intersections",
"Definition:Open Mapping"
] |
proofwiki-3497 | Weak Local Compactness is Preserved under Open Continuous Surjection | Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be topological spaces.
Let $\phi: T_A \to T_B$ be a continuous mapping which is also an open mapping and a surjection.
If $T_A$ is weakly locally compact, then $T_B$ is also weakly locally compact. | Let $\phi$ be a mapping which is surjective, continuous and open.
Let $T_A$ be weakly locally compact.
Take $b \in S_B$.
Let $V$ be a neighbourhood of $b$.
Since $\phi$ is surjective:
:$\forall y \in S_B: \exists x \in S_A: x \in \map {\phi^{-1} } y$
From the weak local compactness of $T_A$ and the continuity of $\phi$... | Let $T_A = \struct {S_A, \tau_A}$ and $T_B = \struct {S_B, \tau_B}$ be [[Definition:Topological Space|topological spaces]].
Let $\phi: T_A \to T_B$ be a [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]] which is also an [[Definition:Open Mapping|open mapping]] and a [[Definition:Surjection|sur... | Let $\phi$ be a [[Definition:Mapping|mapping]] which is [[Definition:Surjection|surjective]], [[Definition:Everywhere Continuous Mapping (Topology)|continuous]] and [[Definition:Open Mapping|open]].
Let $T_A$ be [[Definition:Weakly Locally Compact Space|weakly locally compact]].
Take $b \in S_B$.
Let $V$ be a [[Defi... | Weak Local Compactness is Preserved under Open Continuous Surjection | https://proofwiki.org/wiki/Weak_Local_Compactness_is_Preserved_under_Open_Continuous_Surjection | https://proofwiki.org/wiki/Weak_Local_Compactness_is_Preserved_under_Open_Continuous_Surjection | [
"Weakly Locally Compact Spaces",
"Open Mappings",
"Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Open Mapping",
"Definition:Surjection",
"Definition:Weakly Locally Compact Space",
"Definition:Weakly Locally Compact Space"
] | [
"Definition:Mapping",
"Definition:Surjection",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Open Mapping",
"Definition:Weakly Locally Compact Space",
"Definition:Neighborhood (Topology)/Point",
"Definition:Surjection",
"Definition:Weakly Locally Compact Space",
"Definition:Cont... |
proofwiki-3498 | Compactness Properties Preserved under Projection Mapping | Let $I$ be an indexing set with countable cardinality.
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a family of topological spaces indexed by $I$.
Let $\ds \struct {S, \tau} = \prod_{\alpha \mathop \in I} \struct {S_\alpha, \tau_\alpha}$ be the product space of $\family {\struct {S_\alpha,... | First note that Projection from Product Topology is Continuous.
Also note that Projection is Surjection.
It follows from Compactness Properties Preserved under Continuous Surjection that:
:Compact Space
:$\sigma$-Compact Space
:Countably Compact Space
:Sequentially Compact Space
:Lindelöf Space
are all preserved under ... | Let $I$ be an [[Definition:Indexing Set|indexing set]] with [[Definition:Countable Set|countable cardinality]].
Let $\family {\struct {S_\alpha, \tau_\alpha} }_{\alpha \mathop \in I}$ be a [[Definition:Indexed Family|family]] of [[Definition:Topological Space|topological spaces]] [[Definition:Indexed Family|indexed]] ... | First note that [[Projection from Product Topology is Continuous]].
Also note that [[Projection is Surjection]].
It follows from [[Compactness Properties Preserved under Continuous Surjection]] that:
:[[Definition:Compact Topological Space|Compact Space]]
:[[Definition:Sigma-Compact Space|$\sigma$-Compact Space]]
:[... | Compactness Properties Preserved under Projection Mapping | https://proofwiki.org/wiki/Compactness_Properties_Preserved_under_Projection_Mapping | https://proofwiki.org/wiki/Compactness_Properties_Preserved_under_Projection_Mapping | [
"Compact Topological Spaces",
"Continuous Mappings",
"Projections"
] | [
"Definition:Indexing Set",
"Definition:Countable Set",
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set/Family",
"Definition:Product Space (Topology)",
"Definition:Projection (Mapping Theory)",
"Definition:Compact Topological Space",
"Definition:Compact Topo... | [
"Projection from Product Topology is Continuous",
"Projection is Surjection",
"Compactness Properties Preserved under Continuous Surjection",
"Definition:Compact Topological Space",
"Definition:Sigma-Compact Space",
"Definition:Countably Compact Space",
"Definition:Sequentially Compact Space",
"Defini... |
proofwiki-3499 | Countability Properties Preserved under Projection Mapping | Let $\sequence {\struct {S_\alpha, \tau_\alpha} }$ be a sequence of topological spaces.
Let $\ds \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the product space of $\sequence {\struct {S_\alpha, \tau_\alpha} }$.
Let $\pr_\alpha: \struct {S, \tau} \to \struct {S_\alpha, \tau_\alpha}$ denote the projectio... | First note that Projection from Product Topology is Continuous.
It follows from Continuous Image of Separable Space is Separable that separability is preserved under projections.
Next note that Projection from Product Topology is Open.
It follows from Countability Axioms Preserved under Open Continuous Surjection that:... | Let $\sequence {\struct {S_\alpha, \tau_\alpha} }$ be a [[Definition:Sequence|sequence]] of [[Definition:Topological Space|topological spaces]].
Let $\ds \struct {S, \tau} = \prod \struct {S_\alpha, \tau_\alpha}$ be the [[Definition:Product Space of Topological Spaces|product space]] of $\sequence {\struct {S_\alpha, ... | First note that [[Projection from Product Topology is Continuous]].
It follows from [[Continuous Image of Separable Space is Separable]] that [[Definition:Separable Space|separability]] is preserved under [[Definition:Projection (Mapping Theory)|projections]].
Next note that [[Projection from Product Topology is Ope... | Countability Properties Preserved under Projection Mapping | https://proofwiki.org/wiki/Countability_Properties_Preserved_under_Projection_Mapping | https://proofwiki.org/wiki/Countability_Properties_Preserved_under_Projection_Mapping | [
"Countability Axioms",
"Continuous Mappings",
"Projections"
] | [
"Definition:Sequence",
"Definition:Topological Space",
"Definition:Product Space (Topology)",
"Definition:Projection (Mapping Theory)",
"Definition:Countability Axioms",
"Definition:Separable Space",
"Definition:First-Countable Space",
"Definition:Second-Countable Space"
] | [
"Projection from Product Topology is Continuous",
"Continuous Image of Separable Space is Separable",
"Definition:Separable Space",
"Definition:Projection (Mapping Theory)",
"Projection from Product Topology is Open",
"Countability Axioms Preserved under Open Continuous Surjection",
"Definition:First-Co... |
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