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-4000 | Arens-Fort Space is T1 | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a $T_1$ space. | We have that the Arens-Fort Space is $T_2$.
The result follows from $T_2$ Space is $T_1$ Space.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:T1 Space|$T_1$ space]]. | We have that the [[Arens-Fort Space is T2|Arens-Fort Space is $T_2$]].
The result follows from [[T2 Space is T1 Space|$T_2$ Space is $T_1$ Space]].
{{qed}} | Arens-Fort Space is T1 | https://proofwiki.org/wiki/Arens-Fort_Space_is_T1 | https://proofwiki.org/wiki/Arens-Fort_Space_is_T1 | [
"Arens-Fort Space",
"Examples of T1 Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:T1 Space"
] | [
"Arens-Fort Space is T2",
"T2 Space is T1"
] |
proofwiki-4001 | Arens-Fort Space is T2.5 | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a $T_{2 \frac 1 2}$ space. | We have:
:Fort Space is $T_{2 \frac 1 2}$
:Arens-Fort Space is Expansion of Fort Space.
The result follows from $T_{2 \frac 1 2}$ Property is Preserved under Expansion.
{{qed}}
Category:Arens-Fort Space
Category:Examples of T2.5 Spaces
63vekqpmf9f0cu6tne4djia88gxy7v1 | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]]. | We have:
:[[Fort Space is T2.5|Fort Space is $T_{2 \frac 1 2}$]]
:[[Arens-Fort Space is Expansion of Fort Space]].
The result follows from [[T2.5 Property is Preserved under Expansion|$T_{2 \frac 1 2}$ Property is Preserved under Expansion]].
{{qed}}
[[Category:Arens-Fort Space]]
[[Category:Examples of T2.5 Spaces]]
... | Arens-Fort Space is T2.5 | https://proofwiki.org/wiki/Arens-Fort_Space_is_T2.5 | https://proofwiki.org/wiki/Arens-Fort_Space_is_T2.5 | [
"Arens-Fort Space",
"Examples of T2.5 Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:T2.5 Space"
] | [
"Fort Space is T2.5",
"Arens-Fort Space is Expansion of Countable Fort Space",
"T2.5 Property is Preserved under Expansion",
"Category:Arens-Fort Space",
"Category:Examples of T2.5 Spaces"
] |
proofwiki-4002 | Arens-Fort Space is T5 | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a $T_5$ space. | Let $A, B \subseteq S$ such that $A$ and $B$ are separated.
Let $p = \tuple {0, 0}$.
If $p \notin A$ and $p \notin B$ then $A$ and $B$ are both open and the problem is solved.
Otherwise $p$ must be in exactly one of them, because if $p$ were in both they could not be separated.
{{WLOG}}, suppose $p \in A$.
Then $p \not... | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:T5 Space|$T_5$ space]]. | Let $A, B \subseteq S$ such that $A$ and $B$ are [[Definition:Separated Sets|separated]].
Let $p = \tuple {0, 0}$.
If $p \notin A$ and $p \notin B$ then $A$ and $B$ are both [[Definition:Open Set (Topology)|open]] and the problem is solved.
Otherwise $p$ must be in exactly one of them, because if $p$ were in both t... | Arens-Fort Space is T5 | https://proofwiki.org/wiki/Arens-Fort_Space_is_T5 | https://proofwiki.org/wiki/Arens-Fort_Space_is_T5 | [
"Arens-Fort Space",
"Examples of T5 Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:T5 Space"
] | [
"Definition:Separated Sets",
"Definition:Open Set/Topology",
"Definition:Separated Sets",
"Definition:Open Set/Topology",
"Definition:Arens-Fort Space",
"Set is Subset of its Topological Closure",
"Set Closure is Smallest Closed Set/Topology",
"Definition:Closure (Topology)",
"Definition:Closed Set/... |
proofwiki-4003 | Arens-Fort Space is Completely Normal | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a completely normal space.
Consequently, $T$ satisfies all weaker separation axioms. | We have:
:Arens-Fort Space is $T_1$
:Arens-Fort Space is $T_5$
and so by definition $T$ is completely normal.
{{qed}}
See Sequence of Implications of Separation Axioms for confirmation of the statement about weaker separation axioms. | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:Completely Normal Space|completely normal space]].
Consequently, $T$ satisfies all weaker [[Definition:Separation Axioms|separation axioms]]. | We have:
:[[Arens-Fort Space is T1|Arens-Fort Space is $T_1$]]
:[[Arens-Fort Space is T5|Arens-Fort Space is $T_5$]]
and so by definition $T$ is [[Definition:Completely Normal Space|completely normal]].
{{qed}}
See [[Sequence of Implications of Separation Axioms]] for confirmation of the statement about weaker [[Defi... | Arens-Fort Space is Completely Normal | https://proofwiki.org/wiki/Arens-Fort_Space_is_Completely_Normal | https://proofwiki.org/wiki/Arens-Fort_Space_is_Completely_Normal | [
"Arens-Fort Space",
"Examples of Completely Normal Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Completely Normal Space",
"Definition:Tychonoff Separation Axioms"
] | [
"Arens-Fort Space is T1",
"Arens-Fort Space is T5",
"Definition:Completely Normal Space",
"Sequence of Implications of Separation Axioms",
"Definition:Tychonoff Separation Axioms"
] |
proofwiki-4004 | Arens-Fort Space is not First-Countable | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is not a first-countable space. | {{AimForCont}} $T$ is first-countable.
Then there exists a countable local base $\ds B_0 = \sequence {U_i}_{i \mathop = 1}^\infty$ for $\tuple {0, 0}$.
Suppose there does not exist a point $\tuple {n_i, m_i} \in U_i$ such that $n_i > i$ and $m_i > i$.
Then:
:$\forall \tuple {n, m} \in U_i: n \le i$ or $m \le i$
Now sup... | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is not a [[Definition:First-Countable Space|first-countable space]]. | {{AimForCont}} $T$ is [[Definition:First-Countable Space|first-countable]].
Then there exists a [[Definition:Countable Set|countable]] [[Definition:Local Basis|local base]] $\ds B_0 = \sequence {U_i}_{i \mathop = 1}^\infty$ for $\tuple {0, 0}$.
Suppose there does not exist a point $\tuple {n_i, m_i} \in U_i$ such th... | Arens-Fort Space is not First-Countable | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_First-Countable | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_First-Countable | [
"Arens-Fort Space",
"Examples of First-Countable Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:First-Countable Space"
] | [
"Definition:First-Countable Space",
"Definition:Countable Set",
"Definition:Local Basis",
"Definition:Arens-Fort Space",
"Definition:Infinite Set",
"Definition:Finite Set",
"Definition:Infinite Set",
"Definition:Arens-Fort Space",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhoo... |
proofwiki-4005 | Countable Space is Separable | Let $T = \struct {S, \tau}$ be a topological space where $S$ is a countable set.
Then $T$ is a separable space. | By definition, a topological space $T = \struct {S, \tau}$ is separable if there exists a countable subset of $S$ which is everywhere dense in $T$.
The closure of $S$ in $S$ is trivially $S$.
So, by definition, $S$ is everywhere dense in $S$.
As $S$ is countable by definition, the result follows.
{{qed}}
Category:Separ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] where $S$ is a [[Definition:Countable Set|countable set]].
Then $T$ is a [[Definition:Separable Space|separable space]]. | By definition, a [[Definition:Topological Space|topological space]] $T = \struct {S, \tau}$ is [[Definition:Separable Space|separable]] if there exists a [[Definition:Countable Set|countable]] [[Definition:Subset|subset]] of $S$ which is [[Definition:Everywhere Dense|everywhere dense]] in $T$.
The [[Definition:Closure... | Countable Space is Separable | https://proofwiki.org/wiki/Countable_Space_is_Separable | https://proofwiki.org/wiki/Countable_Space_is_Separable | [
"Separable Spaces",
"Countable Sets"
] | [
"Definition:Topological Space",
"Definition:Countable Set",
"Definition:Separable Space"
] | [
"Definition:Topological Space",
"Definition:Separable Space",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Everywhere Dense",
"Definition:Closure (Topology)",
"Definition:Everywhere Dense",
"Definition:Countable Set",
"Category:Separable Spaces",
"Category:Countable Sets"
] |
proofwiki-4006 | Arens-Fort Space is Separable | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a separable space. | We have that the Arens-Fort space is an expansion of a countable Fort space.
So $S$ is countable.
The result follows from Countable Space is Separable.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:Separable Space|separable space]]. | We have that the [[Definition:Arens-Fort Space|Arens-Fort space]] is an [[Definition:Expansion of Topology|expansion]] of a [[Definition:Countable Fort Space|countable Fort space]].
So $S$ is [[Definition:Countable Set|countable]].
The result follows from [[Countable Space is Separable]].
{{qed}} | Arens-Fort Space is Separable | https://proofwiki.org/wiki/Arens-Fort_Space_is_Separable | https://proofwiki.org/wiki/Arens-Fort_Space_is_Separable | [
"Arens-Fort Space",
"Examples of Separable Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Separable Space"
] | [
"Definition:Arens-Fort Space",
"Definition:Expansion of Topology",
"Definition:Fort Space/Countable",
"Definition:Countable Set",
"Countable Space is Separable"
] |
proofwiki-4007 | Arens-Fort Space is Sigma-Compact | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a $\sigma$-compact space. | From Arens-Fort Space is Countable we have that $S$ is countably infinite.
Let $\tau_d$ be the discrete topology on $S$.
Consider the identity function $I_S: \struct {S, \tau_d} \to \struct {S, \tau}$.
From Mapping from Discrete Space is Continuous, $I_S$ is continuous.
From Identity Mapping is Surjection, $I_S$ is a s... | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:Sigma-Compact Space|$\sigma$-compact space]]. | From [[Arens-Fort Space is Countable]] we have that $S$ is [[Definition:Countably Infinite Set|countably infinite]].
Let $\tau_d$ be the [[Definition:Discrete Space|discrete topology]] on $S$.
Consider the [[Definition:Identity Mapping|identity function]] $I_S: \struct {S, \tau_d} \to \struct {S, \tau}$.
From [[Mapp... | Arens-Fort Space is Sigma-Compact/Proof 1 | https://proofwiki.org/wiki/Arens-Fort_Space_is_Sigma-Compact | https://proofwiki.org/wiki/Arens-Fort_Space_is_Sigma-Compact/Proof_1 | [
"Arens-Fort Space is Sigma-Compact",
"Arens-Fort Space",
"Examples of Sigma-Compact Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Sigma-Compact Space"
] | [
"Arens-Fort Space is Countable",
"Definition:Countably Infinite/Set",
"Definition:Discrete Topology",
"Definition:Identity Mapping",
"Mapping from Discrete Space is Continuous",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Identity Mapping is Surjection",
"Definition:Surjection",
"Countab... |
proofwiki-4008 | Arens-Fort Space is Sigma-Compact | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a $\sigma$-compact space. | The result follows from:
:Arens-Fort Space is Countable
:Countable Space is $\sigma$-Compact.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:Sigma-Compact Space|$\sigma$-compact space]]. | The result follows from:
:[[Arens-Fort Space is Countable]]
:[[Countable Space is Sigma-Compact|Countable Space is $\sigma$-Compact]].
{{qed}} | Arens-Fort Space is Sigma-Compact/Proof 2 | https://proofwiki.org/wiki/Arens-Fort_Space_is_Sigma-Compact | https://proofwiki.org/wiki/Arens-Fort_Space_is_Sigma-Compact/Proof_2 | [
"Arens-Fort Space is Sigma-Compact",
"Arens-Fort Space",
"Examples of Sigma-Compact Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Sigma-Compact Space"
] | [
"Arens-Fort Space is Countable",
"Countable Space is Sigma-Compact"
] |
proofwiki-4009 | Arens-Fort Space is Lindelöf | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a Lindelöf space. | We have:
:The Arens-Fort Space is $\sigma$-Compact.
:A $\sigma$-compact space is Lindelöf Space.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:Lindelöf Space|Lindelöf space]]. | We have:
:The [[Arens-Fort Space is Sigma-Compact|Arens-Fort Space is $\sigma$-Compact]].
:A [[Sigma-Compact Space is Lindelöf Space|$\sigma$-compact space is Lindelöf Space]].
{{qed}} | Arens-Fort Space is Lindelöf | https://proofwiki.org/wiki/Arens-Fort_Space_is_Lindelöf | https://proofwiki.org/wiki/Arens-Fort_Space_is_Lindelöf | [
"Arens-Fort Space",
"Examples of Lindelöf Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Lindelöf Space"
] | [
"Arens-Fort Space is Sigma-Compact",
"Sigma-Compact Space is Lindelöf"
] |
proofwiki-4010 | Arens-Fort Space is not Weakly Locally Compact | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is not a weakly locally compact space. | We have that Neighborhood of Origin in Arens-Fort Space is not Compact.
So $\tuple {0, 0}$ is a point in $S$ which is not contained in a compact neighborhood.
Hence, by definition, $T$ is not weakly locally compact.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is not a [[Definition:Weakly Locally Compact Space|weakly locally compact space]]. | We have that [[Neighborhood of Origin in Arens-Fort Space is not Compact]].
So $\tuple {0, 0}$ is a point in $S$ which is not contained in a [[Definition:Compact Topological Subspace|compact]] [[Definition:Neighborhood (Topology)|neighborhood]].
Hence, by definition, $T$ is not [[Definition:Weakly Locally Compact Spa... | Arens-Fort Space is not Weakly Locally Compact | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_Weakly_Locally_Compact | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_Weakly_Locally_Compact | [
"Arens-Fort Space",
"Examples of Weakly Locally Compact Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Weakly Locally Compact Space"
] | [
"Neighborhood of Origin in Arens-Fort Space is not Compact",
"Definition:Compact Topological Space/Subspace",
"Definition:Neighborhood (Topology)",
"Definition:Weakly Locally Compact Space"
] |
proofwiki-4011 | Arens-Fort Space is not Compact | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is not a compact space. | Follows from:
:Arens-Fort Space is not Weakly Locally Compact
:Compact Space is Weakly Locally Compact.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is not a [[Definition:Compact Topological Space|compact space]]. | Follows from:
:[[Arens-Fort Space is not Weakly Locally Compact]]
:[[Compact Space is Weakly Locally Compact]].
{{qed}} | Arens-Fort Space is not Compact | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_Compact | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_Compact | [
"Arens-Fort Space",
"Examples of Compact Topological Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Compact Topological Space"
] | [
"Arens-Fort Space is not Weakly Locally Compact",
"Compact Space is Weakly Locally Compact"
] |
proofwiki-4012 | Arens-Fort Space is not Countably Compact | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is not a countably compact space. | {{AimForCont}} the Arens-Fort space is countably compact.
From Arens-Fort Space is Lindelöf, it is also Lindelöf.
From Countably Compact Lindelöf Space is Compact, $T$ is compact.
But this contradicts Arens-Fort Space is not Compact.
Hence the result from Proof by Contradiction.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is not a [[Definition:Countably Compact Space|countably compact space]]. | {{AimForCont}} the [[Definition:Arens-Fort Space|Arens-Fort space]] is [[Definition:Countably Compact Space|countably compact]].
From [[Arens-Fort Space is Lindelöf]], it is also [[Definition:Lindelöf Space|Lindelöf]].
From [[Countably Compact Lindelöf Space is Compact]], $T$ is [[Definition:Compact Topological Space... | Arens-Fort Space is not Countably Compact | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_Countably_Compact | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_Countably_Compact | [
"Arens-Fort Space",
"Examples of Countably Compact Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Countably Compact Space"
] | [
"Definition:Arens-Fort Space",
"Definition:Countably Compact Space",
"Arens-Fort Space is Lindelöf",
"Definition:Lindelöf Space",
"Countably Compact Lindelöf Space is Compact",
"Definition:Compact Topological Space",
"Definition:Contradiction",
"Arens-Fort Space is not Compact",
"Proof by Contradict... |
proofwiki-4013 | T3 Lindelöf Space is Paracompact | Let $T = \struct {S, \tau}$ be a $T_3$ space which is also Lindelöf.
Then $T$ is paracompact. | Follows directly from:
{{begin-itemize}}
{{item||$T_3$ Lindelöf Space is Strongly Paracompact}}
{{item||Strongly Paracompact Space is Paracompact}}.
{{end-itemize}}
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:T3 Space|$T_3$ space]] which is also [[Definition:Lindelöf Space|Lindelöf]].
Then $T$ is [[Definition:Paracompact Space|paracompact]]. | Follows directly from:
{{begin-itemize}}
{{item||[[T3 Lindelöf Space is Strongly Paracompact|$T_3$ Lindelöf Space is Strongly Paracompact]]}}
{{item||[[Strongly Paracompact Space is Paracompact]]}}.
{{end-itemize}}
{{qed}} | T3 Lindelöf Space is Paracompact | https://proofwiki.org/wiki/T3_Lindelöf_Space_is_Paracompact | https://proofwiki.org/wiki/T3_Lindelöf_Space_is_Paracompact | [
"T3 Spaces",
"Lindelöf Spaces",
"Paracompact Spaces",
"Sequence of Implications of Compactness Properties in T3 Space"
] | [
"Definition:T3 Space",
"Definition:Lindelöf Space",
"Definition:Paracompact Space"
] | [
"T3 Lindelöf Space is Strongly Paracompact",
"Strongly Paracompact Space is Paracompact"
] |
proofwiki-4014 | Arens-Fort Space is Paracompact | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a paracompact space. | Let $\CC$ be any open cover of $T$.
Let $H \in \CC$ be any open set which contains $\tuple {0, 0}$.
For all $s \in S$ such that $s \ne \tuple {0, 0}$, we have that $\set s$ is open in $T$ by definition of the Arens-Fort space.
So the open cover of $T$ which consists of $H$ together with all the open sets $\set s$, wher... | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:Paracompact Space|paracompact space]]. | Let $\CC$ be any [[Definition:Open Cover|open cover]] of $T$.
Let $H \in \CC$ be any [[Definition:Open Set (Topology)|open set]] which contains $\tuple {0, 0}$.
For all $s \in S$ such that $s \ne \tuple {0, 0}$, we have that $\set s$ is [[Definition:Open Set (Topology)|open]] in $T$ by definition of the [[Definition:... | Arens-Fort Space is Paracompact/Proof 1 | https://proofwiki.org/wiki/Arens-Fort_Space_is_Paracompact | https://proofwiki.org/wiki/Arens-Fort_Space_is_Paracompact/Proof_1 | [
"Arens-Fort Space is Paracompact",
"Arens-Fort Space",
"Examples of Paracompact Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Paracompact Space"
] | [
"Definition:Open Cover",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Arens-Fort Space",
"Definition:Open Cover",
"Definition:Open Set/Topology",
"Definition:Refinement of Cover",
"Definition:Locally Finite Cover",
"Definition:Paracompact Space"
] |
proofwiki-4015 | Arens-Fort Space is Paracompact | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a paracompact space. | We have that:
:Arens-Fort Space is Lindelöf
:Arens-Fort Space is $T_3$.
The result follows from $T_3$ Lindelöf Space is Paracompact.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:Paracompact Space|paracompact space]]. | We have that:
:[[Arens-Fort Space is Lindelöf]]
:[[Arens-Fort Space is T3|Arens-Fort Space is $T_3$]].
The result follows from [[T3 Lindelöf Space is Paracompact|$T_3$ Lindelöf Space is Paracompact]].
{{qed}} | Arens-Fort Space is Paracompact/Proof 2 | https://proofwiki.org/wiki/Arens-Fort_Space_is_Paracompact | https://proofwiki.org/wiki/Arens-Fort_Space_is_Paracompact/Proof_2 | [
"Arens-Fort Space is Paracompact",
"Arens-Fort Space",
"Examples of Paracompact Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Paracompact Space"
] | [
"Arens-Fort Space is Lindelöf",
"Arens-Fort Space is T3",
"T3 Lindelöf Space is Paracompact"
] |
proofwiki-4016 | First-Countability is not Continuous Invariant | Let $T_A = \struct {A, \tau_A}$ and $T_B = \struct {B, \tau_B}$ be topological spaces.
Let $\phi: T_A \to T_B$ be a continuous mapping.
If $T_A$ is a first-countable space, then it does not necessarily follow that $T_B$ is also first-countable. | Let $T_S = \struct {S, \tau_S}$ be the Arens-Fort space.
Let $T_D = \struct {S, \tau_D}$ be the discrete space, also on $S$.
Let $I_S: S \to S$ be the identity mapping on $S$.
From Mapping from Discrete Space is Continuous, we have that $I_S$ is a continuous mapping.
Then we have:
:Discrete Space is First-Countable
:Ar... | Let $T_A = \struct {A, \tau_A}$ and $T_B = \struct {B, \tau_B}$ be [[Definition:Topological Space|topological spaces]].
Let $\phi: T_A \to T_B$ be a [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]].
If $T_A$ is a [[Definition:First-Countable Space|first-countable space]], then it does not n... | Let $T_S = \struct {S, \tau_S}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Let $T_D = \struct {S, \tau_D}$ be the [[Definition:Discrete Space|discrete space]], also on $S$.
Let $I_S: S \to S$ be the [[Definition:Identity Mapping|identity mapping]] on $S$.
From [[Mapping from Discrete Space is Continuo... | First-Countability is not Continuous Invariant | https://proofwiki.org/wiki/First-Countability_is_not_Continuous_Invariant | https://proofwiki.org/wiki/First-Countability_is_not_Continuous_Invariant | [
"First-Countable Spaces",
"Examples of Continuous Invariants"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:First-Countable Space",
"Definition:First-Countable Space"
] | [
"Definition:Arens-Fort Space",
"Definition:Discrete Topology",
"Definition:Identity Mapping",
"Mapping from Discrete Space is Continuous",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Discrete Space is First-Countable",
"Arens-Fort Space is not First-Countable",
"Definition:Continuous Mappi... |
proofwiki-4017 | Second-Countability is not Continuous Invariant | Let $T_A = \struct {A, \tau_A}$ and $T_B = \struct {B, \tau_B}$ be topological spaces.
Let $\phi: T_A \to T_B$ be a continuous mapping.
If $T_A$ is a second-countable space, then it does not necessarily follow that $T_B$ is also second-countable. | Let $T_S = \struct {S, \tau_S}$ be the Arens-Fort space.
Let $T_D = \struct {S, \tau_D}$ be the discrete space, also on $S$.
As $S$ is countable, from Arens-Fort Space is Expansion of Countable Fort Space, it follows that $T_D = \struct {S, \tau_D}$ is a countable discrete space.
Let $I_S: S \to S$ be the identity mapp... | Let $T_A = \struct {A, \tau_A}$ and $T_B = \struct {B, \tau_B}$ be [[Definition:Topological Space|topological spaces]].
Let $\phi: T_A \to T_B$ be a [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]].
If $T_A$ is a [[Definition:Second-Countable Space|second-countable space]], then it does not... | Let $T_S = \struct {S, \tau_S}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Let $T_D = \struct {S, \tau_D}$ be the [[Definition:Discrete Space|discrete space]], also on $S$.
As $S$ is [[Definition:Countable Set|countable]], from [[Arens-Fort Space is Expansion of Countable Fort Space]], it follows that $... | Second-Countability is not Continuous Invariant | https://proofwiki.org/wiki/Second-Countability_is_not_Continuous_Invariant | https://proofwiki.org/wiki/Second-Countability_is_not_Continuous_Invariant | [
"Second-Countable Spaces",
"Examples of Continuous Invariants"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Second-Countable Space",
"Definition:Second-Countable Space"
] | [
"Definition:Arens-Fort Space",
"Definition:Discrete Topology",
"Definition:Countable Set",
"Arens-Fort Space is Expansion of Countable Fort Space",
"Definition:Discrete Topology/Countable",
"Definition:Identity Mapping",
"Mapping from Discrete Space is Continuous",
"Definition:Continuous Mapping (Topo... |
proofwiki-4018 | Arens-Fort Space is not Connected | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is not a connected space. | Consider $p \in S$ such that $p \ne \tuple {0, 0}$.
From Clopen Points in Arens-Fort Space, we have that $\set p$ is both open and closed in $T$.
So by definition of closed set, $\relcomp S {\set p}$ is also both open and closed in $T$.
So, by definition, $\set p \mid \relcomp S {\set p}$ is a separation of $T$
Hence t... | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is not a [[Definition:Connected Topological Space|connected space]]. | Consider $p \in S$ such that $p \ne \tuple {0, 0}$.
From [[Clopen Points in Arens-Fort Space]], we have that $\set p$ is both [[Definition:Open Set (Topology)|open]] and [[Definition:Closed Set (Topology)|closed]] in $T$.
So by definition of [[Definition:Closed Set (Topology)|closed set]], $\relcomp S {\set p}$ is al... | Arens-Fort Space is not Connected | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_Connected | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_Connected | [
"Arens-Fort Space",
"Examples of Connected Topological Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Connected Topological Space"
] | [
"Clopen Points in Arens-Fort Space",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Separation (Topology)",
"Definition:Connected Topological Space"
] |
proofwiki-4019 | Arens-Fort Space is not Locally Connected | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is not a locally connected space. | Let $\UU_0$ be a local basis for $\tuple {0, 0}$.
Let $U \in \UU_0$.
By definition of local basis, $U$ is open in $T$.
From Clopen Points in Arens-Fort Space, $\set {\tuple {0, 0} }$ is not open in $T$.
So $U \ne \set {\tuple {0, 0} }$.
Therefore:
:$\exists p \in U: p \ne \tuple {0, 0}$
From Singleton of Element is Su... | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is not a [[Definition:Locally Connected Space|locally connected space]]. | Let $\UU_0$ be a [[Definition:Local Basis|local basis]] for $\tuple {0, 0}$.
Let $U \in \UU_0$.
By definition of [[Definition:Local Basis|local basis]], $U$ is [[Definition:Open Set (Topology)|open]] in $T$.
From [[Clopen Points in Arens-Fort Space]], $\set {\tuple {0, 0} }$ is not [[Definition:Open Set (Topology)|o... | Arens-Fort Space is not Locally Connected | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_Locally_Connected | https://proofwiki.org/wiki/Arens-Fort_Space_is_not_Locally_Connected | [
"Arens-Fort Space",
"Examples of Locally Connected Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Locally Connected Space"
] | [
"Definition:Local Basis",
"Definition:Local Basis",
"Definition:Open Set/Topology",
"Clopen Points in Arens-Fort Space",
"Definition:Open Set/Topology",
"Singleton of Element is Subset",
"Clopen Points in Arens-Fort Space",
"Definition:Clopen Set",
"Definition:Local Basis",
"Connected iff no Prope... |
proofwiki-4020 | Clopen Points in Arens-Fort Space | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Let $q \in S: q \ne \tuple {0, 0}$.
Then $\set q$ is both open and closed in $T$.
$\set {\tuple {0, 0} }$ itself is closed, but not open. | We have that $\set q$ is finite so $\relcomp S {\set q}$ is cofinite.
So $\relcomp S {\set q}$ is open and so $\set q$ is closed.
Then we have that $\tuple {0, 0} \notin \set q$ so $\set q$ is open.
However, $\tuple {0, 0} \notin \relcomp S {\set {\tuple {0, 0} } }$ and $\relcomp S {\set {\tuple {0, 0} } }$ is clearly ... | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Let $q \in S: q \ne \tuple {0, 0}$.
Then $\set q$ is both [[Definition:Open Set (Topology)|open]] and [[Definition:Closed Set (Topology)|closed]] in $T$.
$\set {\tuple {0, 0} }$ itself is [[Definition:Closed Set (Topology)|closed]... | We have that $\set q$ is [[Definition:Finite Set|finite]] so $\relcomp S {\set q}$ is [[Definition:Cofinite Subset|cofinite]].
So $\relcomp S {\set q}$ is [[Definition:Open Set (Topology)|open]] and so $\set q$ is [[Definition:Closed Set (Topology)|closed]].
Then we have that $\tuple {0, 0} \notin \set q$ so $\set q... | Clopen Points in Arens-Fort Space | https://proofwiki.org/wiki/Clopen_Points_in_Arens-Fort_Space | https://proofwiki.org/wiki/Clopen_Points_in_Arens-Fort_Space | [
"Arens-Fort Space",
"Examples of Clopen Sets"
] | [
"Definition:Arens-Fort Space",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology"
] | [
"Definition:Finite Set",
"Definition:Cofinite Subset",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Finite Set",
"Definition:Cofinite Subset",
"Definition:Closed Set/Topology"
] |
proofwiki-4021 | Neighborhood of Origin of Arens-Fort Space is Closed | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Every neighborhood of $\tuple {0, 0}$ is closed in $T$. | Let $H \subseteq S$ such that:
:$\exists U \in \tau: \tuple {0, 0} \in U \subseteq H \subseteq S$
that is: such that $H$ is a neighborhood of $\tuple {0, 0}$ in $T$.
As $\tuple {0, 0} \in H$ it follows that $\tuple {0, 0} \notin \relcomp S H$.
So, by definition of the Arens-Fort space, $\relcomp S H$ is open in $T$.
So... | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Every [[Definition:Neighborhood of Point|neighborhood]] of $\tuple {0, 0}$ is [[Definition:Closed Set (Topology)|closed]] in $T$. | Let $H \subseteq S$ such that:
:$\exists U \in \tau: \tuple {0, 0} \in U \subseteq H \subseteq S$
that is: such that $H$ is a [[Definition:Neighborhood of Point|neighborhood]] of $\tuple {0, 0}$ in $T$.
As $\tuple {0, 0} \in H$ it follows that $\tuple {0, 0} \notin \relcomp S H$.
So, by definition of the [[Definitio... | Neighborhood of Origin of Arens-Fort Space is Closed | https://proofwiki.org/wiki/Neighborhood_of_Origin_of_Arens-Fort_Space_is_Closed | https://proofwiki.org/wiki/Neighborhood_of_Origin_of_Arens-Fort_Space_is_Closed | [
"Arens-Fort Space",
"Examples of Closed Sets"
] | [
"Definition:Arens-Fort Space",
"Definition:Neighborhood (Topology)/Point",
"Definition:Closed Set/Topology"
] | [
"Definition:Neighborhood (Topology)/Point",
"Definition:Arens-Fort Space",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology"
] |
proofwiki-4022 | Arens-Fort Space is Zero Dimensional | Let $T = \struct {S, \tau}$ denote the Arens-Fort space.
Then $T$ is zero dimensional. | Let $q \in S$ such that $q \ne \tuple {0, 0}$.
Then from Clopen Points in Arens-Fort Space, $\set q$ is clopen.
So $\forall q \in S, q \ne \tuple {0, 0}: \set {\set q}$ is a local basis for $q$.
If we take the neighborhoods of $\tuple {0, 0}$ that are open we get that they are also closed from Neighborhood of Origin of... | Let $T = \struct {S, \tau}$ denote the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is [[Definition:Zero Dimensional Space|zero dimensional]]. | Let $q \in S$ such that $q \ne \tuple {0, 0}$.
Then from [[Clopen Points in Arens-Fort Space]], $\set q$ is [[Definition:Clopen Set|clopen]].
So $\forall q \in S, q \ne \tuple {0, 0}: \set {\set q}$ is a [[Definition:Local Basis|local basis]] for $q$.
If we take the [[Definition:Neighborhood of Point|neighborhoods]... | Arens-Fort Space is Zero Dimensional | https://proofwiki.org/wiki/Arens-Fort_Space_is_Zero_Dimensional | https://proofwiki.org/wiki/Arens-Fort_Space_is_Zero_Dimensional | [
"Arens-Fort Space",
"Examples of Zero Dimensional Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Zero Dimensional Space"
] | [
"Clopen Points in Arens-Fort Space",
"Definition:Clopen Set",
"Definition:Local Basis",
"Definition:Neighborhood (Topology)/Point",
"Definition:Open Neighborhood/Point",
"Definition:Closed Set/Topology",
"Neighborhood of Origin of Arens-Fort Space is Closed",
"Definition:Set Union",
"Definition:Loca... |
proofwiki-4023 | Arens-Fort Space is Totally Separated | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is totally separated. | We have that:
:The Arens-Fort Space is Zero Dimensional.
:The Arens-Fort Space is $T_0$.
Then we have that a Zero Dimensional $T_0$ Space is Totally Separated.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is [[Definition:Totally Separated Space|totally separated]]. | We have that:
:The [[Arens-Fort Space is Zero Dimensional]].
:The [[Arens-Fort Space is T0|Arens-Fort Space is $T_0$]].
Then we have that a [[Zero Dimensional T0 Space is Totally Separated|Zero Dimensional $T_0$ Space is Totally Separated]].
{{qed}} | Arens-Fort Space is Totally Separated | https://proofwiki.org/wiki/Arens-Fort_Space_is_Totally_Separated | https://proofwiki.org/wiki/Arens-Fort_Space_is_Totally_Separated | [
"Arens-Fort Space",
"Examples of Totally Separated Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Totally Separated Space"
] | [
"Arens-Fort Space is Zero Dimensional",
"Arens-Fort Space is T0",
"Zero Dimensional T0 Space is Totally Separated"
] |
proofwiki-4024 | Isolated Points in Arens-Fort Space | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Let $q \in S: q \ne \tuple {0, 0}$.
Then $q$ is an isolated point of $T$. | If $q \ne \tuple {0, 0}$ then from Clopen Points in Arens-Fort Space we have that $\set q$ is both closed and open in $T$.
In particular, $\set q$ is open in $T$.
The result follows from Point in Topological Space is Open iff Isolated.
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Let $q \in S: q \ne \tuple {0, 0}$.
Then $q$ is an [[Definition:Isolated Point (Topology)|isolated point]] of $T$. | If $q \ne \tuple {0, 0}$ then from [[Clopen Points in Arens-Fort Space]] we have that $\set q$ is both [[Definition:Closed Set (Topology)|closed]] and [[Definition:Open Set (Topology)|open]] in $T$.
In particular, $\set q$ is [[Definition:Open Set (Topology)|open]] in $T$.
The result follows from [[Point in Topologic... | Isolated Points in Arens-Fort Space | https://proofwiki.org/wiki/Isolated_Points_in_Arens-Fort_Space | https://proofwiki.org/wiki/Isolated_Points_in_Arens-Fort_Space | [
"Arens-Fort Space",
"Examples of Isolated Points"
] | [
"Definition:Arens-Fort Space",
"Definition:Isolated Point (Topology)"
] | [
"Clopen Points in Arens-Fort Space",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Point in Topological Space is Open iff Isolated"
] |
proofwiki-4025 | Point in Topological Space is Open iff Isolated | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $x \in S$.
Then $\left\{{x}\right\}$ is open in $T$ {{iff}} $x$ is an isolated point of $T$. | Let $\left\{{x}\right\}$ be open in $T$.
Then we have that:
:$\exists \left\{{x}\right\} \in \tau: x \in \left\{{x}\right\}\subseteq S$
This is precisely the condition which ensures that $x$ is an isolated point of $T$.
Now suppose that $x$ is an isolated point of $T$.
Then by definition there exists a open set of $T$ ... | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Then $\left\{{x}\right\}$ is [[Definition:Open Set (Topology)|open]] in $T$ {{iff}} $x$ is an [[Definition:Isolated Point (Topology)|isolated point]] of $T$. | Let $\left\{{x}\right\}$ be [[Definition:Open Set (Topology)|open]] in $T$.
Then we have that:
:$\exists \left\{{x}\right\} \in \tau: x \in \left\{{x}\right\}\subseteq S$
This is precisely the condition which ensures that $x$ is an [[Definition:Isolated Point (Topology)|isolated point]] of $T$.
Now suppose that $x$... | Point in Topological Space is Open iff Isolated | https://proofwiki.org/wiki/Point_in_Topological_Space_is_Open_iff_Isolated | https://proofwiki.org/wiki/Point_in_Topological_Space_is_Open_iff_Isolated | [
"Open Sets",
"Isolated Points"
] | [
"Definition:Topological Space",
"Definition:Open Set/Topology",
"Definition:Isolated Point (Topology)"
] | [
"Definition:Open Set/Topology",
"Definition:Isolated Point (Topology)",
"Definition:Isolated Point (Topology)",
"Definition:Open Set/Topology",
"Definition:Isolated Point (Topology)",
"Definition:Open Set/Topology",
"Category:Open Sets",
"Category:Isolated Points"
] |
proofwiki-4026 | Arens-Fort Space is Scattered | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Then $T$ is a scattered space. | Let $H \subseteq T$ such that $H \ne \set {\tuple {0, 0} }$.
Then $\exists x \in H: x \ne \tuple {0, 0}$.
From Clopen Points in Arens-Fort Space, every point of $T$ apart from $\tuple {0, 0}$ is open in $T$.
So $\set x$ is an open set of $T$.
So $H \cap \set x = \set x$ and so $x$ is isolated in $H$.
Thus $H$ contains ... | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Then $T$ is a [[Definition:Scattered Space|scattered space]]. | Let $H \subseteq T$ such that $H \ne \set {\tuple {0, 0} }$.
Then $\exists x \in H: x \ne \tuple {0, 0}$.
From [[Clopen Points in Arens-Fort Space]], every point of $T$ apart from $\tuple {0, 0}$ is [[Definition:Open Point|open]] in $T$.
So $\set x$ is an [[Definition:Open Set (Topology)|open set]] of $T$.
So $H \c... | Arens-Fort Space is Scattered | https://proofwiki.org/wiki/Arens-Fort_Space_is_Scattered | https://proofwiki.org/wiki/Arens-Fort_Space_is_Scattered | [
"Arens-Fort Space",
"Examples of Scattered Spaces"
] | [
"Definition:Arens-Fort Space",
"Definition:Scattered Space"
] | [
"Clopen Points in Arens-Fort Space",
"Definition:Open Point",
"Definition:Open Set/Topology",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Isolated Point (Topology)/Subset",
"Singleton Point is Isolated",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Isolated Point (Topology... |
proofwiki-4027 | Fortissimo Space is not Metrizable | Let $T = \struct {S, \tau_p}$ be a Fortissimo space.
Then $\tau_p$ is not a metrizable topology. | From:
:Metric Space is First-Countable
:Fortissimo Space is not First-Countable
it is deduced that $\tau_p$ is not a metrizable topology.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fortissimo Space|Fortissimo space]].
Then $\tau_p$ is not a [[Definition:Metrizable Topology|metrizable topology]]. | From:
:[[Metric Space is First-Countable]]
:[[Fortissimo Space is not First-Countable]]
it is deduced that $\tau_p$ is not a [[Definition:Metrizable Topology|metrizable topology]].
{{qed}} | Fortissimo Space is not Metrizable | https://proofwiki.org/wiki/Fortissimo_Space_is_not_Metrizable | https://proofwiki.org/wiki/Fortissimo_Space_is_not_Metrizable | [
"Fortissimo Spaces",
"Examples of Metrizable Spaces"
] | [
"Definition:Fortissimo Space",
"Definition:Metrizable Space"
] | [
"Metric Space is First-Countable",
"Fortissimo Space is not First-Countable",
"Definition:Metrizable Space"
] |
proofwiki-4028 | Fortissimo Space is Paracompact | Let $T = \struct {S, \tau_p}$ be a Fortissimo space.
Then $T$ is a paracompact space. | Let $\CC$ be an open cover of $T$.
Let $U_p \in \CC$ be an open set in that cover which contains $p$.
All the points of $S \setminus U_p$ are open points by definition of Fortissimo space.
Then $\DD = \set {\set s: s \in S \setminus U_p} \cup \set {U_p}$ is an open refinement of $\CC$.
Take $k \in S$:
:If $k = p$, then... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fortissimo Space|Fortissimo space]].
Then $T$ is a [[Definition:Paracompact Space|paracompact space]]. | Let $\CC$ be an [[Definition:Open Cover|open cover]] of $T$.
Let $U_p \in \CC$ be an [[Definition:Open Set (Topology)|open set]] in that cover which contains $p$.
All the points of $S \setminus U_p$ are [[Definition:Open Point|open points]] by definition of [[Definition:Fortissimo Space|Fortissimo space]].
Then $\DD... | Fortissimo Space is Paracompact | https://proofwiki.org/wiki/Fortissimo_Space_is_Paracompact | https://proofwiki.org/wiki/Fortissimo_Space_is_Paracompact | [
"Fortissimo Spaces",
"Examples of Paracompact Spaces"
] | [
"Definition:Fortissimo Space",
"Definition:Paracompact Space"
] | [
"Definition:Open Cover",
"Definition:Open Set/Topology",
"Definition:Open Point",
"Definition:Fortissimo Space",
"Definition:Open Refinement",
"Definition:Neighborhood (Topology)/Point",
"Definition:Set Intersection",
"Definition:Subset",
"Definition:Neighborhood (Topology)/Point",
"Definition:Set... |
proofwiki-4029 | Clopen Points in Modified Fort Space | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then all points in $S \setminus \set {a, b}$ are clopen in $T$.
$a$ and $b$ themselves are not open in $T$, but they are closed in $T$. | Let $p \in S: p \notin \set {a, b}$.
From the definition of modified Fort space, any subset of $S \setminus \set {a, b}$ is open in $T$.
It follows directly that as $p \in S \setminus \set {a, b}$ we have that $\set p \subseteq S \setminus \set {a, b}$.
Hence $p$ is open in $T$.
As for $a$ and $b$, we have that $S \set... | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then all points in $S \setminus \set {a, b}$ are [[Definition:Clopen Point|clopen]] in $T$.
$a$ and $b$ themselves are not [[Definition:Open Point|open]] in $T$, but they are [[Definition:Closed Point|closed]] in $T$. | Let $p \in S: p \notin \set {a, b}$.
From the definition of [[Definition:Modified Fort Space|modified Fort space]], any subset of $S \setminus \set {a, b}$ is [[Definition:Open Set (Topology)|open]] in $T$.
It follows directly that as $p \in S \setminus \set {a, b}$ we have that $\set p \subseteq S \setminus \set {a,... | Clopen Points in Modified Fort Space | https://proofwiki.org/wiki/Clopen_Points_in_Modified_Fort_Space | https://proofwiki.org/wiki/Clopen_Points_in_Modified_Fort_Space | [
"Modified Fort Spaces",
"Examples of Clopen Points"
] | [
"Definition:Modified Fort Space",
"Definition:Clopen Set/Clopen Point",
"Definition:Open Point",
"Definition:Closed Point"
] | [
"Definition:Modified Fort Space",
"Definition:Open Set/Topology",
"Definition:Open Point",
"Definition:Finite Set",
"Definition:Open Point",
"Definition:Finite Set",
"Definition:Cofinite Subset",
"Definition:Open Set/Topology",
"Definition:Closed Point"
] |
proofwiki-4030 | Modified Fort Space is Compact | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is a compact space. | Let $\CC$ be an open cover of $T$.
Then $\exists U \in \CC: a \in U$.
Then by definition of modified Fort space, $U$ is cofinite.
That is, $S \setminus U$ is finite.
Then $S \setminus U$ is covered by a finite subcover of $\CC$.
Hence, by definition, $T$ is compact.
{{qed}} | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is a [[Definition:Compact Topological Space|compact space]]. | Let $\CC$ be an [[Definition:Open Cover|open cover]] of $T$.
Then $\exists U \in \CC: a \in U$.
Then by definition of [[Definition:Modified Fort Space|modified Fort space]], $U$ is [[Definition:Cofinite Subset|cofinite]].
That is, $S \setminus U$ is [[Definition:Finite Set|finite]].
Then $S \setminus U$ is covered ... | Modified Fort Space is Compact | https://proofwiki.org/wiki/Modified_Fort_Space_is_Compact | https://proofwiki.org/wiki/Modified_Fort_Space_is_Compact | [
"Modified Fort Spaces",
"Examples of Compact Topological Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:Compact Topological Space"
] | [
"Definition:Open Cover",
"Definition:Modified Fort Space",
"Definition:Cofinite Subset",
"Definition:Finite Set",
"Definition:Subcover/Finite",
"Definition:Compact Topological Space"
] |
proofwiki-4031 | Modified Fort Space is T1 | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is a $T_1$ space. | Let $p, q \in S: p \ne q$.
There are two options:
{{begin-itemize}}
{{item|(1):|At least one of $p$ and $q$ is not in $\set {a, b}$}}
{{item|(2):|Both $p$ and $q$ are in $\set {a, b}$.}}
{{end-itemize}}
;Proof by Cases
{{begin-itemize}}
{{item|(1):|First let $p \notin \set {a, b}$ or $q \notin \set {a, b}$.
{{WLOG}}, l... | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is a [[Definition:T1 Space|$T_1$ space]]. | Let $p, q \in S: p \ne q$.
There are two options:
{{begin-itemize}}
{{item|(1):|At least one of $p$ and $q$ is not in $\set {a, b}$}}
{{item|(2):|Both $p$ and $q$ are in $\set {a, b}$.}}
{{end-itemize}}
;[[Proof by Cases]]
{{begin-itemize}}
{{item|(1):|First let $p \notin \set {a, b}$ or $q \notin \set {a, b}$.
{{W... | Modified Fort Space is T1 | https://proofwiki.org/wiki/Modified_Fort_Space_is_T1 | https://proofwiki.org/wiki/Modified_Fort_Space_is_T1 | [
"Modified Fort Spaces",
"Examples of T1 Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:T1 Space"
] | [
"Proof by Cases",
"Clopen Points in Modified Fort Space",
"Definition:Finite Set",
"Definition:Open Set/Topology",
"Definition:Finite Set",
"Definition:Open Set/Topology",
"Definition:T1 Space"
] |
proofwiki-4032 | Modified Fort Space is not T2 | Let $T = \struct {S, \tau_{a, b} }$ be a modified Fort space.
Then $T$ is not a $T_2$ (Hausdorff) space. | Consider $U, V \in \tau_{a, b}$ such that $a \in U, b \in V$.
We have that both $U$ and $V$ are cofinite.
So $U$ and $V$ must be infinite.
Suppose $U \cap V = \O$.
Then from Intersection with Complement is Empty iff Subset it follows that $U \subseteq \relcomp S V$ and so $U$ is finite.
But this contradicts the fact th... | Let $T = \struct {S, \tau_{a, b} }$ be a [[Definition:Modified Fort Space|modified Fort space]].
Then $T$ is not a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | Consider $U, V \in \tau_{a, b}$ such that $a \in U, b \in V$.
We have that both $U$ and $V$ are [[Definition:Cofinite Subset|cofinite]].
So $U$ and $V$ must be [[Definition:Infinite Set|infinite]].
Suppose $U \cap V = \O$.
Then from [[Intersection with Complement is Empty iff Subset]] it follows that $U \subseteq \... | Modified Fort Space is not T2 | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_T2 | https://proofwiki.org/wiki/Modified_Fort_Space_is_not_T2 | [
"Modified Fort Spaces",
"Examples of Hausdorff Spaces"
] | [
"Definition:Modified Fort Space",
"Definition:T2 Space"
] | [
"Definition:Cofinite Subset",
"Definition:Infinite Set",
"Intersection with Complement is Empty iff Subset",
"Definition:Finite Set",
"Definition:Infinite Set",
"Definition:T2 Space"
] |
proofwiki-4033 | Basis for Euclidean Topology on Real Number Line | Let $\R$ be the set of real numbers.
Let $\BB$ be the set of subsets of $\R$ defined as:
:$\BB = \set {\openint a b: a, b \in \R}$
That is, $\BB$ is the set of all open real intervals of $\R$:
:$\openint a b := \set {x \in \R: a < x < b}$
Then $\BB$ forms a basis for the Euclidean topology on $\R$. | From Real Number Line is Metric Space, one can define an open interval on the set of real numbers in terms of an $\epsilon$-neighborhood.
From Open Real Interval is Open Ball, an open interval $\openint a b$ is the open $\epsilon$-ball $\map {B_\epsilon} \alpha$.
Then from Metric Induces Topology we have that:
:$\BB = ... | Let $\R$ be the [[Definition:Real Number|set of real numbers]].
Let $\BB$ be the [[Definition:Set of Sets|set]] of [[Definition:Subset|subsets]] of $\R$ defined as:
:$\BB = \set {\openint a b: a, b \in \R}$
That is, $\BB$ is the set of all [[Definition:Open Real Interval|open real intervals]] of $\R$:
:$\openint a b ... | From [[Real Number Line is Metric Space]], one can define an [[Definition:Open Real Interval|open interval]] on the [[Definition:Real Number|set of real numbers]] in terms of an [[Definition:Epsilon-Neighborhood (Real Number Line)|$\epsilon$-neighborhood]].
From [[Open Real Interval is Open Ball]], an [[Definition:Ope... | Basis for Euclidean Topology on Real Number Line | https://proofwiki.org/wiki/Basis_for_Euclidean_Topology_on_Real_Number_Line | https://proofwiki.org/wiki/Basis_for_Euclidean_Topology_on_Real_Number_Line | [
"Real Number Line with Euclidean Topology",
"Examples of Topological Bases"
] | [
"Definition:Real Number",
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Real Interval/Open",
"Definition:Basis (Topology)",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line"
] | [
"Real Number Line is Metric Space",
"Definition:Real Interval/Open",
"Definition:Real Number",
"Definition:Neighborhood (Real Analysis)/Epsilon",
"Open Real Interval is Open Ball",
"Definition:Real Interval/Open",
"Definition:Open Ball",
"Metric Induces Topology",
"Definition:Topology"
] |
proofwiki-4034 | Real Number Line satisfies all Separation Axioms | Let $\struct {\R, \tau_d}$ be the the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ fulfils all separation axioms:
=== Real Number Line with Euclidean Topology is $T_0$ ===
{{:Real Number Line with Euclidean Topology is T0}}
=== Real Number Line with Euclidean Topology is $T_1$ ===
{... | * Real Number Line is Metric Space
* Metric Space fulfils all Separation Axioms
{{qed}} | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|the real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ fulfils all [[Definition:Separation Axioms|separation axioms]]:
=== [[Real Number Line with Euclidean Topology is T0|Real Number Line with Eu... | * [[Real Number Line is Metric Space]]
* [[Metric Space fulfils all Separation Axioms]]
{{qed}} | Real Number Line satisfies all Separation Axioms | https://proofwiki.org/wiki/Real_Number_Line_satisfies_all_Separation_Axioms | https://proofwiki.org/wiki/Real_Number_Line_satisfies_all_Separation_Axioms | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of Separation Axioms"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Tychonoff Separation Axioms",
"Real Number Line with Euclidean Topology is T0",
"Real Number Line with Euclidean Topology is T1",
"Real Number Line with Euclidean Topology is T2",
"Real Number Line with Euclidean Topology is Sem... | [
"Real Number Line is Metric Space",
"Metric Space fulfils all Separation Axioms"
] |
proofwiki-4035 | Elementary Amalgamation Theorem | Let $\MM$ and $\NN$ be $\LL$-structures.
Let $B$ be a subset of the universe of $\MM$ such that there is a partial elementary embedding $f: B \to \NN$.
There is an elementary extension $\AA$ of $\MM$ and an elementary embedding $g: \NN \to \AA$ such that $\map g {\map f b} = b$ for all $b \in B$.
:<nowiki>$\array {
& &... | The proof is essentially a straightforward application of the Compactness Theorem on the collection of sentences with parameters true in each of $\MM$ and $\NN$, but it requires some tedious set-up in handling symbols of the language involved to make sure that we get the map $g$ with the properties we want in the end.
... | Let $\MM$ and $\NN$ be $\LL$-[[Definition:First-Order Structure|structures]].
Let $B$ be a [[Definition:Subset|subset]] of the universe of $\MM$ such that there is a [[Definition:Partial Elementary Embedding|partial elementary embedding]] $f: B \to \NN$.
There is an [[Definition:Elementary Extension|elementary exten... | The proof is essentially a straightforward application of the [[Compactness Theorem]] on the collection of sentences with parameters true in each of $\MM$ and $\NN$, but it requires some tedious set-up in handling symbols of the language involved to make sure that we get the map $g$ with the properties we want in the e... | Elementary Amalgamation Theorem | https://proofwiki.org/wiki/Elementary_Amalgamation_Theorem | https://proofwiki.org/wiki/Elementary_Amalgamation_Theorem | [
"Model Theory for Predicate Logic"
] | [
"Definition:Structure for Predicate Logic",
"Definition:Subset",
"Definition:Elementary Embedding/Partial Elementary Embedding",
"Definition:Elementary Extension",
"Definition:Elementary Embedding"
] | [
"Compactness Theorem",
"Definition:Set",
"Definition:Logical Language",
"Definition:Logical Language",
"Definition:Elementary Diagram",
"Definition:Theory",
"Definition:Model (Logic)",
"Compactness Theorem",
"Definition:Ordered Tuple",
"Compactness Theorem",
"Definition:Elementary Embedding",
... |
proofwiki-4036 | Real Number Line is Non-Meager | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is non-meager. | We have that the Real Number Line is Complete Metric Space.
From the Baire Category Theorem, a complete metric space is also a Baire space.
The result follows from Baire Space is Non-Meager.
{{qed}}
{{ADC|Baire Category Theorem|3}} | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ is [[Definition:Non-Meager Space|non-meager]]. | We have that the [[Real Number Line is Complete Metric Space]].
From the [[Baire Category Theorem]], a [[Definition:Complete Metric Space|complete metric space]] is also a [[Definition:Baire Space (Topology)|Baire space]].
The result follows from [[Baire Space is Non-Meager]].
{{qed}}
{{ADC|Baire Category Theorem|3}... | Real Number Line is Non-Meager/Proof 1 | https://proofwiki.org/wiki/Real_Number_Line_is_Non-Meager | https://proofwiki.org/wiki/Real_Number_Line_is_Non-Meager/Proof_1 | [
"Real Number Line is Non-Meager",
"Real Number Line with Euclidean Topology",
"Examples of Non-Meager Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Meager Space/Non-Meager"
] | [
"Real Number Line is Complete Metric Space",
"Baire Category Theorem",
"Definition:Complete Metric Space",
"Definition:Baire Space (Topology)",
"Baire Space is Non-Meager"
] |
proofwiki-4037 | Real Number Line is Non-Meager | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is non-meager. | This proof does not use the {{Axiom-link|Dependent Choice}}, as it uses intrinsic properties of the real numbers that do not necessarily hold for the general complete metric space.
{{ProofWanted}} | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ is [[Definition:Non-Meager Space|non-meager]]. | This proof does not use the {{Axiom-link|Dependent Choice}}, as it uses intrinsic properties of the [[Definition:Real Number|real numbers]] that do not necessarily hold for the general [[Definition:Complete Metric Space|complete metric space]].
{{ProofWanted}} | Real Number Line is Non-Meager/Proof 2 | https://proofwiki.org/wiki/Real_Number_Line_is_Non-Meager | https://proofwiki.org/wiki/Real_Number_Line_is_Non-Meager/Proof_2 | [
"Real Number Line is Non-Meager",
"Real Number Line with Euclidean Topology",
"Examples of Non-Meager Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Meager Space/Non-Meager"
] | [
"Definition:Real Number",
"Definition:Complete Metric Space"
] |
proofwiki-4038 | Baire Space iff Open Sets are Non-Meager | Let $T = \struct {S, \tau}$ be a topological space.
Then $T$ is a Baire space {{iff}} every non-empty open set of $T$ is non-meager in $T$. | We prove the contrapositive:
:$T$ is not a Baire space {{iff}} there exists a non-empty open set of $T$ which is meager in $T$.
We have by definition of Baire space:
:$T$ is a Baire space {{iff}} the interior of the union of any countable set of closed sets of $T$ which are nowhere dense is empty.
Therefore:
:$T$ is no... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then $T$ is a [[Definition:Baire Space (Topology)|Baire space]] {{iff}} every [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Set (Topology)|open]] set of $T$ is [[Definition:Non-Meager Space|non-meager]] in $T$. | We prove the [[Definition:Contrapositive Statement|contrapositive]]:
:$T$ is not a [[Definition:Baire Space (Topology)|Baire space]] {{iff}} there exists a [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Set (Topology)|open]] set of $T$ which is [[Definition:Meager Space|meager]] in $T$.
We have by [[Definit... | Baire Space iff Open Sets are Non-Meager | https://proofwiki.org/wiki/Baire_Space_iff_Open_Sets_are_Non-Meager | https://proofwiki.org/wiki/Baire_Space_iff_Open_Sets_are_Non-Meager | [
"Baire Spaces",
"Non-Meager Spaces"
] | [
"Definition:Topological Space",
"Definition:Baire Space (Topology)",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Definition:Meager Space/Non-Meager"
] | [
"Definition:Contrapositive Statement",
"Definition:Baire Space (Topology)",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Definition:Meager Space",
"Definition:Baire Space (Topology)/Definition 3",
"Definition:Interior (Topology)",
"Definition:Set Union",
"Definition:Countable Set",
... |
proofwiki-4039 | Extension Realizing All Types | Let $\MM$ be an $\LL$-structure.
Let $M$ be its universe.
There is an elementary extension $\NN$ of $\MM$ such that every type over $M$ (relative to $\MM$) is realized in $\NN$. | Let $\map S \MM$ denote the set containing all complete types over $M$ of every number of free variables.
Let $\kappa = \size{\map S \MM}$.
Use a bijection between $\kappa$ and $\map S \MM$ to write the elements of $\map S \MM$ as $p_\alpha$ for $\alpha < \kappa$.
For each $\alpha < \kappa$, let $N_\alpha$ be an elemen... | Let $\MM$ be an $\LL$-[[Definition:First-Order Structure|structure]].
Let $M$ be its [[Definition:Universe (Model Theory)|universe]].
There is an [[Definition:Elementary Extension|elementary extension]] $\NN$ of $\MM$ such that every [[Definition:Type|type]] over $M$ (relative to $\MM$) is [[Definition:Realization o... | Let $\map S \MM$ denote the [[Definition:Set|set]] containing all [[Definition:Complete Type|complete types]] over $M$ of every number of [[Definition:Free Variable|free variables]].
Let $\kappa = \size{\map S \MM}$.
Use a [[Definition:Bijection|bijection]] between $\kappa$ and $\map S \MM$ to write the [[Definition... | Extension Realizing All Types | https://proofwiki.org/wiki/Extension_Realizing_All_Types | https://proofwiki.org/wiki/Extension_Realizing_All_Types | [
"Model Theory for Predicate Logic"
] | [
"Definition:Structure for Predicate Logic",
"Definition:Universe (Model Theory)",
"Definition:Elementary Extension",
"Definition:Type",
"Definition:Type"
] | [
"Definition:Set",
"Definition:Type",
"Definition:Free Variable",
"Definition:Bijection",
"Definition:Element",
"Definition:Elementary Extension",
"Definition:Ordered Tuple",
"Definition:Type",
"Definition:Elementary Extension",
"Definition:Ordered Tuple",
"Type is Realized in some Elementary Ext... |
proofwiki-4040 | Baire Space is Non-Meager | Let $T = \struct {S, \tau}$ be a Baire space (in the context of topology).
Then $T$ is non-meager in $T$. | From Baire Space iff Open Sets are Non-Meager, all open sets of $T$ are non-meager in $T$.
But $T$ itself is an open set of $T$ by definition of topological space.
Hence the result.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Baire Space (Topology)|Baire space]] (in the context of [[Definition:Topology (Mathematical Branch)|topology]]).
Then $T$ is [[Definition:Non-Meager Space|non-meager]] in $T$. | From [[Baire Space iff Open Sets are Non-Meager]], all [[Definition:Open Set (Topology)|open sets]] of $T$ are [[Definition:Non-Meager Space|non-meager]] in $T$.
But $T$ itself is an [[Definition:Open Set (Topology)|open set]] of $T$ by definition of [[Definition:Topological Space|topological space]].
Hence the resul... | Baire Space is Non-Meager | https://proofwiki.org/wiki/Baire_Space_is_Non-Meager | https://proofwiki.org/wiki/Baire_Space_is_Non-Meager | [
"Baire Spaces",
"Non-Meager Spaces"
] | [
"Definition:Baire Space (Topology)",
"Definition:Topology (Mathematical Branch)",
"Definition:Meager Space/Non-Meager"
] | [
"Baire Space iff Open Sets are Non-Meager",
"Definition:Open Set/Topology",
"Definition:Meager Space/Non-Meager",
"Definition:Open Set/Topology",
"Definition:Topological Space"
] |
proofwiki-4041 | Real Number Line is Separable | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is a separable space. | {{Recall|Separable Space|separable space}}
{{:Definition:Separable Space}}
From Rational Numbers form Subset of Real Numbers, the set of rational numbers $\Q$ forms a subset of the set of real numbers $\R$.
We have:
:Rational Numbers are Countably Infinite
:Rationals are Everywhere Dense in Topological Space of Reals
T... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ is a [[Definition:Separable Space|separable space]]. | {{Recall|Separable Space|separable space}}
{{:Definition:Separable Space}}
From [[Rational Numbers form Subset of Real Numbers]], the [[Definition:Rational Numbers|set of rational numbers $\Q$]] forms a [[Definition:Subset|subset]] of the [[Definition:Real Numbers|set of real numbers $\R$]].
We have:
:[[Rational Numb... | Real Number Line is Separable/Proof 1 | https://proofwiki.org/wiki/Real_Number_Line_is_Separable | https://proofwiki.org/wiki/Real_Number_Line_is_Separable/Proof_1 | [
"Real Number Line is Separable",
"Real Number Line with Euclidean Topology",
"Examples of Separable Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Separable Space"
] | [
"Rational Numbers form Subset of Real Numbers",
"Definition:Rational Number",
"Definition:Subset",
"Definition:Real Number",
"Rational Numbers are Countably Infinite",
"Rational Numbers are Everywhere Dense in Set of Real Numbers/Topology",
"Definition:Separable Space"
] |
proofwiki-4042 | Real Number Line is Separable | Let $\struct {\R, \tau_d}$ be the real number line with the usual (Euclidean) topology.
Then $\struct {\R, \tau_d}$ is a separable space. | Follows from:
:Real Number Line is Second-Countable
:Second-Countable Space is Separable
{{qed}} | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line with the usual (Euclidean) topology]].
Then $\struct {\R, \tau_d}$ is a [[Definition:Separable Space|separable space]]. | Follows from:
:[[Real Number Line is Second-Countable]]
:[[Second-Countable Space is Separable]]
{{qed}} | Real Number Line is Separable/Proof 2 | https://proofwiki.org/wiki/Real_Number_Line_is_Separable | https://proofwiki.org/wiki/Real_Number_Line_is_Separable/Proof_2 | [
"Real Number Line is Separable",
"Real Number Line with Euclidean Topology",
"Examples of Separable Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Separable Space"
] | [
"Real Number Line is Second-Countable",
"Second-Countable Space is Separable"
] |
proofwiki-4043 | Double Pointed Space is T3 iff Factor Space is T3 | Let $T_S = \struct {S, \tau_S}$ be a topological space.
Let $D = \struct {A, \set {\O, A} }$ be the indiscrete space on an arbitrary doubleton $A = \set {a, b}$.
Let $T = \struct {T_S \times D, \tau}$ be the double pointed topological space on $T_S$.
Then $T$ is a $T_3$ space {{iff}} $T_S$ is also a $T_3$ space. | Let $S' = S \times \set {a, b}$.
Let $F' \subseteq S'$ such that $F'$ is closed in $T$.
Then there exists $F \subseteq S$ such that:
:$F' = F \times \set {a, b}$
or:
:$F' = F \times \O$ by definition of the double pointed topology.
If $F' = F \times \O$ then $F' = \O$ from Cartesian Product is Empty iff Factor is Empty... | Let $T_S = \struct {S, \tau_S}$ be a [[Definition:Topological Space|topological space]].
Let $D = \struct {A, \set {\O, A} }$ be the [[Definition:Indiscrete Space|indiscrete space]] on an [[Definition:Arbitrary|arbitrary]] [[Definition:Doubleton|doubleton]] $A = \set {a, b}$.
Let $T = \struct {T_S \times D, \tau}$ be... | Let $S' = S \times \set {a, b}$.
Let $F' \subseteq S'$ such that $F'$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
Then there exists $F \subseteq S$ such that:
:$F' = F \times \set {a, b}$
or:
:$F' = F \times \O$ by definition of the [[Definition:Double Pointed Topology|double pointed topology]].
If $F' = ... | Double Pointed Space is T3 iff Factor Space is T3 | https://proofwiki.org/wiki/Double_Pointed_Space_is_T3_iff_Factor_Space_is_T3 | https://proofwiki.org/wiki/Double_Pointed_Space_is_T3_iff_Factor_Space_is_T3 | [
"Separation Axioms on Double Pointed Topology",
"Examples of T3 Spaces"
] | [
"Definition:Topological Space",
"Definition:Indiscrete Topology",
"Definition:Arbitrary",
"Definition:Doubleton",
"Definition:Double Pointed Topology",
"Definition:T3 Space",
"Definition:T3 Space"
] | [
"Definition:Closed Set/Topology",
"Definition:Double Pointed Topology",
"Cartesian Product is Empty iff Factor is Empty",
"Open and Closed Sets in Multiple Pointed Topology",
"Definition:Closed Set/Topology",
"Definition:T3 Space",
"Definition:T3 Space",
"Definition:T3 Space",
"Definition:Indiscrete... |
proofwiki-4044 | Double Pointed Space is T4 iff Factor Space is T4 | Let $T_S = \struct {S, \tau_S}$ be a topological space.
Let $D = \struct {A, \set {\O, A} }$ be the indiscrete space on an arbitrary doubleton $A = \set {a, b}$.
Let $T = \struct {T_S \times D, \tau}$ be the double pointed topological space on $T_S$.
Then $T$ is a $T_4$ space {{iff}} $T_S$ is also a $T_4$ space. | Let $S' = S \times \set {a, b}$.
Let $H' \subseteq S'$ such that $H$ is closed in $T$.
Then $H' = H \times \set {a, b}$ or $H' = H \times \O$ by definition of the double pointed topology.
If $H' = H \times \O$ then $H' = \O$ from Cartesian Product is Empty iff Factor is Empty, and the result is trivial.
So suppose $H' ... | Let $T_S = \struct {S, \tau_S}$ be a [[Definition:Topological Space|topological space]].
Let $D = \struct {A, \set {\O, A} }$ be the [[Definition:Indiscrete Space|indiscrete space]] on an [[Definition:Arbitrary|arbitrary]] [[Definition:Doubleton|doubleton]] $A = \set {a, b}$.
Let $T = \struct {T_S \times D, \tau}$ be... | Let $S' = S \times \set {a, b}$.
Let $H' \subseteq S'$ such that $H$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
Then $H' = H \times \set {a, b}$ or $H' = H \times \O$ by definition of the [[Definition:Double Pointed Topology|double pointed topology]].
If $H' = H \times \O$ then $H' = \O$ from [[Cartesian... | Double Pointed Space is T4 iff Factor Space is T4 | https://proofwiki.org/wiki/Double_Pointed_Space_is_T4_iff_Factor_Space_is_T4 | https://proofwiki.org/wiki/Double_Pointed_Space_is_T4_iff_Factor_Space_is_T4 | [
"Separation Axioms on Double Pointed Topology",
"Examples of T4 Spaces"
] | [
"Definition:Topological Space",
"Definition:Indiscrete Topology",
"Definition:Arbitrary",
"Definition:Doubleton",
"Definition:Double Pointed Topology",
"Definition:T4 Space",
"Definition:T4 Space"
] | [
"Definition:Closed Set/Topology",
"Definition:Double Pointed Topology",
"Cartesian Product is Empty iff Factor is Empty",
"Open and Closed Sets in Multiple Pointed Topology",
"Definition:Closed Set/Topology",
"Definition:T4 Space",
"Definition:T4 Space",
"Definition:T4 Space",
"Definition:Indiscrete... |
proofwiki-4045 | Double Pointed Space is T5 iff Factor Space is T5 | Let $T_S = \struct {S, \tau_S}$ be a topological space.
Let $D = \struct {A, \set {\O, A} }$ be the indiscrete space on an arbitrary doubleton $A = \set {a, b}$.
Let $T = \struct {T_S \times D, \tau}$ be the double pointed topological space on $T_S$.
Then $T$ is a $T_5$ space {{iff}} $T_S$ is also a $T_5$ space. | === Necessary Condition ===
Suppose that $T \times D$ is a $T_5$ space.
Let $A, B \subseteq T$ be two separated sets.
First, we will show that $A \times D$ and $B \times D$ are also separated.
To this end, observe that:
{{begin-eqn}}
{{eqn | l = \paren {A \times D}^- \cap \paren {B \times D}
| r = \paren {A^- \ti... | Let $T_S = \struct {S, \tau_S}$ be a [[Definition:Topological Space|topological space]].
Let $D = \struct {A, \set {\O, A} }$ be the [[Definition:Indiscrete Space|indiscrete space]] on an [[Definition:Arbitrary|arbitrary]] [[Definition:Doubleton|doubleton]] $A = \set {a, b}$.
Let $T = \struct {T_S \times D, \tau}$ be... | === Necessary Condition ===
Suppose that $T \times D$ is a [[Definition:T5 Space|$T_5$ space]].
Let $A, B \subseteq T$ be two [[Definition:Separated Sets|separated sets]].
First, we will show that $A \times D$ and $B \times D$ are also [[Definition:Separated Sets|separated]].
To this end, observe that:
{{begin-eq... | Double Pointed Space is T5 iff Factor Space is T5 | https://proofwiki.org/wiki/Double_Pointed_Space_is_T5_iff_Factor_Space_is_T5 | https://proofwiki.org/wiki/Double_Pointed_Space_is_T5_iff_Factor_Space_is_T5 | [
"Separation Axioms on Double Pointed Topology",
"Examples of T5 Spaces"
] | [
"Definition:Topological Space",
"Definition:Indiscrete Topology",
"Definition:Arbitrary",
"Definition:Doubleton",
"Definition:Double Pointed Topology",
"Definition:T5 Space",
"Definition:T5 Space"
] | [
"Definition:T5 Space",
"Definition:Separated Sets",
"Definition:Separated Sets",
"Closure of Subset of Double Pointed Topological Space",
"Cartesian Product of Intersections/Corollary 1",
"Definition:Separated Sets",
"Cartesian Product is Empty iff Factor is Empty",
"Definition:Separated Sets",
"Def... |
proofwiki-4046 | Double Pointed Space is T3.5 iff Factor Space is T3.5 | Let $T_S = \struct {S, \tau_S}$ be a topological space.
Let $D = \struct {A, \set {\O, A} }$ be the indiscrete space on an arbitrary doubleton $A = \set {a, b}$.
Let $T = \struct {T_S \times D, \tau}$ be the double pointed topological space on $T_S$.
Then $T$ is a $T_{3 \frac 1 2}$ space {{iff}} $T_S$ is also a $T_{3 \... | {{ProofWanted}}
Category:Separation Axioms on Double Pointed Topology
Category:Examples of T3.5 Spaces
en4im77omirlmtwyexq8fo752gni9p4 | Let $T_S = \struct {S, \tau_S}$ be a [[Definition:Topological Space|topological space]].
Let $D = \struct {A, \set {\O, A} }$ be the [[Definition:Indiscrete Space|indiscrete space]] on an [[Definition:Arbitrary|arbitrary]] [[Definition:Doubleton|doubleton]] $A = \set {a, b}$.
Let $T = \struct {T_S \times D, \tau}$ be... | {{ProofWanted}}
[[Category:Separation Axioms on Double Pointed Topology]]
[[Category:Examples of T3.5 Spaces]]
en4im77omirlmtwyexq8fo752gni9p4 | Double Pointed Space is T3.5 iff Factor Space is T3.5 | https://proofwiki.org/wiki/Double_Pointed_Space_is_T3.5_iff_Factor_Space_is_T3.5 | https://proofwiki.org/wiki/Double_Pointed_Space_is_T3.5_iff_Factor_Space_is_T3.5 | [
"Separation Axioms on Double Pointed Topology",
"Examples of T3.5 Spaces"
] | [
"Definition:Topological Space",
"Definition:Indiscrete Topology",
"Definition:Arbitrary",
"Definition:Doubleton",
"Definition:Double Pointed Topology",
"Definition:T3.5 Space",
"Definition:T3.5 Space"
] | [
"Category:Separation Axioms on Double Pointed Topology",
"Category:Examples of T3.5 Spaces"
] |
proofwiki-4047 | Square of Vandermonde Matrix | The square of the Vandermonde matrix of order $n$:
: $\mathbf V = \begin{bmatrix}
x_1 & x_2 & \cdots & x_n \\
x_1^2 & x_2^2 & \cdots & x_n^2 \\
\vdots & \vdots & \ddots & \vdots \\
x_1^n & x_2^n & \cdots & x_n^n
\end{bmatrix}$
is symmetrical in $x_1, \ldots, x_n$.
{{questionable|The case $n {{=}} 2$ left me cluel... | {{proof wanted}}
Category:Vandermonde Matrices
5cznpmmrrl7g7g1lwc0sq7m0imj6y3t | The [[Definition:Square (Algebra)|square]] of the [[Definition:Vandermonde Matrix/Formulation 2|Vandermonde matrix of order $n$]]:
: $\mathbf V = \begin{bmatrix}
x_1 & x_2 & \cdots & x_n \\
x_1^2 & x_2^2 & \cdots & x_n^2 \\
\vdots & \vdots & \ddots & \vdots \\
x_1^n & x_2^n & \cdots & x_n^n
\end{bmatrix}$
is sy... | {{proof wanted}}
[[Category:Vandermonde Matrices]]
5cznpmmrrl7g7g1lwc0sq7m0imj6y3t | Square of Vandermonde Matrix | https://proofwiki.org/wiki/Square_of_Vandermonde_Matrix | https://proofwiki.org/wiki/Square_of_Vandermonde_Matrix | [
"Vandermonde Matrices"
] | [
"Definition:Square/Function",
"Definition:Vandermonde Matrix/Formulation 2"
] | [
"Category:Vandermonde Matrices"
] |
proofwiki-4048 | Open and Closed Sets in Multiple Pointed Topology | Let $T = \struct {S, \tau}$ be a topological space.
Let $A$ be a finite set whose cardinality is greater than $1$.
Let $D = \struct {A, \set {\O, A} }$ be the indiscrete space on $A$.
Let $T \times D$ be a multiple pointed topological space generated from $T$ and $D$.
Let $H \subseteq T$.
Then:
:$H \times A$ is open in... | By definition of multiple pointed topology, $T \times D$ is the product space of $T$ with $D$:
:$T \times D = \struct {S \times A, \tau}$
where $\tau$ is the product topology on $S \times A$.
Consider the set $H \times A$.
We have that:
:$H \times A = \pr_1^{-1} \sqbrk H$
where $\pr_1: T \times D \to T$ is the first pr... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A$ be a [[Definition:Finite Set|finite set]] whose [[Definition:Cardinality|cardinality]] is greater than $1$.
Let $D = \struct {A, \set {\O, A} }$ be the [[Definition:Indiscrete Space|indiscrete space]] on $A$.
Let $T \times ... | By definition of [[Definition:Multiple Pointed Topology|multiple pointed topology]], $T \times D$ is the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] of $T$ with $D$:
:$T \times D = \struct {S \times A, \tau}$
where $\tau$ is the [[Definition:Product Topology on Two Factor Spaces|product t... | Open and Closed Sets in Multiple Pointed Topology | https://proofwiki.org/wiki/Open_and_Closed_Sets_in_Multiple_Pointed_Topology | https://proofwiki.org/wiki/Open_and_Closed_Sets_in_Multiple_Pointed_Topology | [
"Multiple Pointed Topology",
"Open Sets",
"Closed Sets",
"Clopen Sets"
] | [
"Definition:Topological Space",
"Definition:Finite Set",
"Definition:Cardinality",
"Definition:Indiscrete Topology",
"Definition:Multiple Pointed Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definiti... | [
"Definition:Multiple Pointed Topology",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Product Topology/Two Factor Spaces",
"Definition:Projection (Mapping Theory)/First Projection",
"Definition:Product Topology/Two Factor Spaces",
"Definition:Continuous Mapping (Topology)",
"Defin... |
proofwiki-4049 | Basis for Box Topology | Let $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be an $I$-indexed family of topological spaces.
Let $S$ be the cartesian product of $\family {S_i}_{i \mathop \in I}$.
That is:
:$\ds S := \prod_{i \mathop \in I} S_i$
Define:
:$\ds \BB := \set {\prod_{i \mathop \in I} U_i: \forall i \in I: U_i \in \tau_i}$
Then ... | Let us check the two conditions for $\BB$ to be a synthetic basis in turn. | Let $\family {\struct {S_i, \tau_i} }_{i \mathop \in I}$ be an [[Definition:Indexed Family|$I$-indexed family]] of [[Definition:Topological Space|topological spaces]].
Let $S$ be the [[Definition:Cartesian Product of Family|cartesian product]] of $\family {S_i}_{i \mathop \in I}$.
That is:
:$\ds S := \prod_{i \mathop... | Let us check the two conditions for $\BB$ to be a [[Definition:Synthetic Basis|synthetic basis]] in turn. | Basis for Box Topology | https://proofwiki.org/wiki/Basis_for_Box_Topology | https://proofwiki.org/wiki/Basis_for_Box_Topology | [
"Box Topology",
"Topological Bases"
] | [
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Cartesian Product/Family of Sets",
"Definition:Basis (Topology)/Synthetic Basis"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Basis (Topology)/Synthetic Basis"
] |
proofwiki-4050 | Space with Open Point is Non-Meager | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$ be an open point.
Then $T$ is a non-meager space. | {{Recall|Meager Space|meager}}
{{:Definition:Meager Space}}
{{Recall|Non-Meager Space|non-meager}}
{{:Definition:Non-Meager Space}}
Let $x \in S$ be an open point of $T$.
That is:
:$\set x \in \tau$
{{AimForCont}} that $T$ is meager.
Let:
:$\ds T = \bigcup \SS$
where $\SS$ is a countable set of subsets of $S$ which are... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$ be an [[Definition:Open Point|open point]].
Then $T$ is a [[Definition:Non-Meager Space|non-meager space]]. | {{Recall|Meager Space|meager}}
{{:Definition:Meager Space}}
{{Recall|Non-Meager Space|non-meager}}
{{:Definition:Non-Meager Space}}
Let $x \in S$ be an [[Definition:Open Point|open point]] of $T$.
That is:
:$\set x \in \tau$
{{AimForCont}} that $T$ is [[Definition:Meager Space|meager]].
Let:
:$\ds T = \bigcup \SS... | Space with Open Point is Non-Meager | https://proofwiki.org/wiki/Space_with_Open_Point_is_Non-Meager | https://proofwiki.org/wiki/Space_with_Open_Point_is_Non-Meager | [
"Non-Meager Spaces"
] | [
"Definition:Topological Space",
"Definition:Open Point",
"Definition:Meager Space/Non-Meager"
] | [
"Definition:Open Point",
"Definition:Meager Space",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Nowhere Dense",
"Definition:Nowhere Dense/Definition 2",
"Set is Subset of its Topological Closure",
"Subset Relation is Transitive",
"Definition:Open Set/Topology",
"Definition:Nowhere... |
proofwiki-4051 | Trivial Topological Space is Non-Meager | Let $T = \struct {S, \tau}$ be a trivial topological space.
Then $T$ is non-meager. | As $T$ is a trivial topological space, by definition $S$ is a singleton: $S = \set s$, say.
Then $\set s$ is an open set.
That is, $s$ is an open point.
The result follows from Space with Open Point is Non-Meager.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Trivial Topological Space|trivial topological space]].
Then $T$ is [[Definition:Non-Meager Space|non-meager]]. | As $T$ is a [[Definition:Trivial Topological Space|trivial topological space]], by definition $S$ is a [[Definition:Singleton|singleton]]: $S = \set s$, say.
Then $\set s$ is an [[Definition:Open Set (Topology)|open set]].
That is, $s$ is an [[Definition:Open Point|open point]].
The result follows from [[Space with ... | Trivial Topological Space is Non-Meager | https://proofwiki.org/wiki/Trivial_Topological_Space_is_Non-Meager | https://proofwiki.org/wiki/Trivial_Topological_Space_is_Non-Meager | [
"Trivial Topological Spaces",
"Examples of Non-Meager Spaces"
] | [
"Definition:Trivial Topological Space",
"Definition:Meager Space/Non-Meager"
] | [
"Definition:Trivial Topological Space",
"Definition:Singleton",
"Definition:Open Set/Topology",
"Definition:Open Point",
"Space with Open Point is Non-Meager"
] |
proofwiki-4052 | Equivalence of Definitions of T2 Space | {{TFAE|def = T2 Space|view = $T_2$ (Hausdorff) space}}
Let $T = \struct {S, \tau}$ be a topological space. | === Definition 1 implies Definition 2 ===
Let $T = \struct {S, \tau}$ be a topological space for which:
:$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \O$
Let us take any arbitrary $x, y \in S: x \ne y$.
Let $\CC_x$ be the set of all closed neighborhoods of $x$:
:$\CC_x = \set {H: \... | {{TFAE|def = T2 Space|view = $T_2$ (Hausdorff) space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]]. | === Definition 1 implies Definition 2 ===
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] for which:
:$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \O$
Let us take any arbitrary $x, y \in S: x \ne y$.
Let $\CC_x$ be the set of all [[Definitio... | Equivalence of Definitions of T2 Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_T2_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_T2_Space | [
"Hausdorff Spaces"
] | [
"Definition:Topological Space"
] | [
"Definition:Topological Space",
"Definition:Closed Neighborhood",
"Definition:Relative Complement",
"Definition:Set Intersection",
"Definition:Closed Neighborhood",
"Definition:Closed Neighborhood",
"Definition:Set Intersection",
"Definition:Closed Neighborhood",
"Definition:Topological Space",
"D... |
proofwiki-4053 | Equivalence of Definitions of T3 Space | {{TFAE|def = T3 Space|view = $T_3$ space}}
Let $T = \struct {S, \tau}$ be a topological space. | === Definition by Open Sets implies Definition by Closed Neighborhoods ===
Let $T = \struct {S, \tau}$ be a topological space for which:
:$\forall F \subseteq S: \relcomp S F \in \tau, y \in \relcomp S F: \exists U, V \in \tau: F \subseteq U, y \in V: U \cap V = \O$
Let $U \in \tau$, and let $x \in U$.
Then by Relativ... | {{TFAE|def = T3 Space|view = $T_3$ space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]]. | === Definition by Open Sets implies Definition by Closed Neighborhoods ===
Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] for which:
:$\forall F \subseteq S: \relcomp S F \in \tau, y \in \relcomp S F: \exists U, V \in \tau: F \subseteq U, y \in V: U \cap V = \O$
Let $U \in \tau... | Equivalence of Definitions of T3 Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_T3_Space | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_T3_Space | [
"T3 Spaces"
] | [
"Definition:Topological Space"
] | [
"Definition:Topological Space",
"Relative Complement of Relative Complement",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Empty Intersection iff Subset of Complement",
"Set Complement inverts Subsets",
"Relative Complement of Relative Complement",
"Definition:Closed Neighborho... |
proofwiki-4054 | Big Implies Saturated | Let $\MM$ be an $\LL$-structure.
Let $\kappa$ be a cardinal.
If $\MM$ is $\kappa$-big, then it is $\kappa$-saturated. | The idea of the proof is:
:to go to some elementary equivalent structure where a type is realized
and:
:to interpret a new relation symbol as the singleton containing some realization of the type.
This lets us write sentences about the extension saying that:
:there must exist an element satisfying the relation
and:
:ev... | Let $\MM$ be an $\LL$-[[Definition:First-Order Structure|structure]].
Let $\kappa$ be a [[Definition:Cardinal|cardinal]].
If $\MM$ is $\kappa$-[[Definition:Big Model|big]], then it is $\kappa$-[[Definition:Saturated Model|saturated]]. | The idea of the proof is:
:to go to some elementary equivalent structure where a type is [[Definition:Realization of Type|realized]]
and:
:to interpret a new relation symbol as the singleton containing some [[Definition:Realization of Type|realization]] of the type.
This lets us write sentences about the extension say... | Big Implies Saturated | https://proofwiki.org/wiki/Big_Implies_Saturated | https://proofwiki.org/wiki/Big_Implies_Saturated | [
"Model Theory for Predicate Logic"
] | [
"Definition:Structure for Predicate Logic",
"Definition:Cardinal",
"Definition:Big Model",
"Definition:Saturated Model"
] | [
"Definition:Type",
"Definition:Type",
"Definition:Type",
"Definition:Cardinality",
"Definition:Type",
"Definition:Logical Language",
"Definition:Constant Symbol",
"Definition:Classes of WFFs/Sentence",
"Definition:Type",
"Definition:Model (Logic)",
"Definition:Ordered Tuple",
"Definition:Type"... |
proofwiki-4055 | Limit Points in T1 Space | Let $T = \struct {S, \tau}$ be a $T_1$ space.
Let $H \subset S$ be any subset of $S$.
Let $x \in H$.
Then $x$ is a limit point of $H$ {{iff}} every neighborhood of $x$ contains infinitely many points of $H$. | === Necessary Condition ===
Suppose every neighborhood of $x$ contains infinitely many points of $H$.
Let $N$ be an open neighborhood of $x$.
Then $H \cap N$ is infinite, and so is $H \cap \paren {N \setminus \set x}$.
Since $N$ is arbitrary, $x$ is a limit point of $H$ by definition.
{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:T1 Space|$T_1$ space]].
Let $H \subset S$ be any [[Definition:Subset|subset]] of $S$.
Let $x \in H$.
Then $x$ is a [[Definition:Limit Point of Set|limit point]] of $H$ {{iff}} every [[Definition:Neighborhood of Point|neighborhood]] of $x$ contains [[Definition:Infinite ... | === Necessary Condition ===
Suppose every [[Definition:Neighborhood of Point|neighborhood]] of $x$ contains [[Definition:Infinite Set|infinitely many]] points of $H$.
Let $N$ be an [[Definition:Open Neighborhood of Point|open neighborhood]] of $x$.
Then $H \cap N$ is [[Definition:Infinite Set|infinite]], and so is $... | Limit Points in T1 Space | https://proofwiki.org/wiki/Limit_Points_in_T1_Space | https://proofwiki.org/wiki/Limit_Points_in_T1_Space | [
"Examples of Limit Points",
"T1 Spaces"
] | [
"Definition:T1 Space",
"Definition:Subset",
"Definition:Limit Point/Topology/Set",
"Definition:Neighborhood (Topology)/Point",
"Definition:Infinite Set"
] | [
"Definition:Neighborhood (Topology)/Point",
"Definition:Infinite Set",
"Definition:Open Neighborhood/Point",
"Definition:Infinite Set",
"Definition:Limit Point/Topology/Set",
"Definition:Neighborhood (Topology)/Point",
"Definition:Open Neighborhood/Point",
"Definition:Limit Point/Topology/Set"
] |
proofwiki-4056 | Singleton Point is Isolated | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
Then $x$ is an isolated point of the singleton set $\set x$, but not necessarily an isolated point of $T$. | Let $U \in \tau$ be an open set of $T$ such that $x \in T$.
The fact that such a $U$ exists follows from the fact that:
:from {{Open-set-axiom|3}}, $S$ is open in $T$
:$x \in S$.
Hence:
:$\set x \subseteq S$
and so from Intersection with Subset is Subset:
$\set x \cap U = \set x$
So by definition, $x$ is an isolated po... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Then $x$ is an [[Definition:Isolated Point (Topology)|isolated point]] of the [[Definition:Singleton|singleton set]] $\set x$, but not necessarily an [[Definition:Isolated Point (Topology)|isolated point of $T$]]. | Let $U \in \tau$ be an [[Definition:Open Set (Topology)|open set]] of $T$ such that $x \in T$.
The fact that such a $U$ exists follows from the fact that:
:from {{Open-set-axiom|3}}, $S$ is open in $T$
:$x \in S$.
Hence:
:$\set x \subseteq S$
and so from [[Intersection with Subset is Subset]]:
$\set x \cap U = \set ... | Singleton Point is Isolated | https://proofwiki.org/wiki/Singleton_Point_is_Isolated | https://proofwiki.org/wiki/Singleton_Point_is_Isolated | [
"Isolated Points",
"Singletons"
] | [
"Definition:Topological Space",
"Definition:Isolated Point (Topology)",
"Definition:Singleton",
"Definition:Isolated Point (Topology)"
] | [
"Definition:Open Set/Topology",
"Intersection with Subset is Subset",
"Definition:Isolated Point (Topology)",
"Topological Space is Discrete iff All Points are Isolated",
"Definition:Discrete Topology",
"Definition:Isolated Point (Topology)",
"Category:Isolated Points",
"Category:Singletons"
] |
proofwiki-4057 | Singleton Set is not Dense-in-itself | Let $T = \struct {S, \tau}$ be a topological space.
Let $x \in S$.
Then the singleton set $\set x$ is not dense-in-itself. | {{Recall|Dense-in-itself|dense-in-itself}}
{{:Definition:Dense-in-itself}}
From Singleton Point is Isolated, $x$ is isolated in $\set x$.
So by definition $\set x$ is not dense-in-itself.
{{qed}}
Category:Dense-in-itself
Category:Singletons
70m0ywinp90b6v7zyolicpff9l7xa3b | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $x \in S$.
Then the [[Definition:Singleton|singleton set]] $\set x$ is not [[Definition:Dense-in-itself|dense-in-itself]]. | {{Recall|Dense-in-itself|dense-in-itself}}
{{:Definition:Dense-in-itself}}
From [[Singleton Point is Isolated]], $x$ is [[Definition:Isolated Point (Topology)|isolated]] in $\set x$.
So by definition $\set x$ is not [[Definition:Dense-in-itself|dense-in-itself]].
{{qed}}
[[Category:Dense-in-itself]]
[[Category:Singl... | Singleton Set is not Dense-in-itself | https://proofwiki.org/wiki/Singleton_Set_is_not_Dense-in-itself | https://proofwiki.org/wiki/Singleton_Set_is_not_Dense-in-itself | [
"Dense-in-itself",
"Singletons"
] | [
"Definition:Topological Space",
"Definition:Singleton",
"Definition:Dense-in-itself"
] | [
"Singleton Point is Isolated",
"Definition:Isolated Point (Topology)",
"Definition:Dense-in-itself",
"Category:Dense-in-itself",
"Category:Singletons"
] |
proofwiki-4058 | Tietze Extension Theorem | Let $T = \struct {S, \tau}$ be a topological space which is normal.
Let $A \subseteq S$ be a closed set in $T$.
Let $f: A \to \R$ be a continuous mapping from $A \subseteq S$ to the real number line under the usual (Euclidean) topology.
Then there exists a continuous extension $g: S \to \R$, that is, such that:
:$\fora... | === Lemma ===
{{:Tietze Extension Theorem/Lemma}}{{qed|lemma}}
First suppose that for any continuous mapping on a closed set there is a continuous extension.
Let $C$ and $D$ be disjoint sets which are closed in $S$.
Define $f: C \cup D \to \R$ by:
:<nowiki>$\map f x = \begin {cases}
0 & : x \in C \\
1 & : x \in D \end ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Normal Space|normal]].
Let $A \subseteq S$ be a [[Definition:Closed Set (Topology)|closed set]] in $T$.
Let $f: A \to \R$ be a [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]] from $A \s... | === [[Tietze Extension Theorem/Lemma|Lemma]] ===
{{:Tietze Extension Theorem/Lemma}}{{qed|lemma}}
First suppose that for any [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]] on a [[Definition:Closed Set (Topology)|closed set]] there is a [[Definition:Continuous Extension|continuous extension... | Tietze Extension Theorem | https://proofwiki.org/wiki/Tietze_Extension_Theorem | https://proofwiki.org/wiki/Tietze_Extension_Theorem | [
"Tietze Extension Theorem",
"Normal Spaces",
"Continuous Mappings"
] | [
"Definition:Topological Space",
"Definition:Normal Space",
"Definition:Closed Set/Topology",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Real Number/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Continuous Extension"
] | [
"Tietze Extension Theorem/Lemma",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Closed Set/Topology",
"Definition:Continuous Extension",
"Definition:Disjoint Sets",
"Definition:Closed Set/Topology",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Continuous Mapp... |
proofwiki-4059 | Hermitian Operator has Orthogonal Eigenvectors | The eigenvectors of a Hermitian operator are orthogonal. | This proof uses Dirac notation.
Note that the inner product is linear in the second argument.
Let $V$ be an inner product space over $\C$.
Let $\hat H$ be a Hermitian operator on $V$.
Let $\lambda_i$ and $\lambda_j$ be distinct eigenvalues of $\hat H$.
Let $\ket {x_i}$ and $\ket {x_j}$ be the eigenvectors associated wi... | The [[Definition:Eigenvector of Linear Operator|eigenvectors]] of a [[Definition:Hermitian Operator|Hermitian operator]] are [[Definition:Orthogonal (Linear Algebra)|orthogonal]]. | This proof uses [[Definition:Dirac Notation|Dirac notation]].
Note that the [[Definition:Inner Product|inner product]] is [[Definition:Linear Transformation|linear]] in the second argument.
Let $V$ be an [[Definition:Inner Product Space|inner product space]] over $\C$.
Let $\hat H$ be a [[Definition:Hermitian Opera... | Hermitian Operator has Orthogonal Eigenvectors/Proof 1 | https://proofwiki.org/wiki/Hermitian_Operator_has_Orthogonal_Eigenvectors | https://proofwiki.org/wiki/Hermitian_Operator_has_Orthogonal_Eigenvectors/Proof_1 | [
"Hermitian Operators",
"Eigenvectors of Linear Operators",
"Hermitian Operator has Orthogonal Eigenvectors"
] | [
"Definition:Eigenvector/Linear Operator",
"Definition:Hermitian Operator",
"Definition:Orthogonal (Linear Algebra)"
] | [
"Definition:Dirac Notation",
"Definition:Inner Product",
"Definition:Linear Transformation",
"Definition:Inner Product Space",
"Definition:Hermitian Operator",
"Definition:Eigenvalue/Linear Operator",
"Definition:Eigenvector/Linear Operator",
"Inner Product is Sesquilinear",
"Inner Product is Sesqui... |
proofwiki-4060 | Hermitian Operator has Orthogonal Eigenvectors | The eigenvectors of a Hermitian operator are orthogonal. | Follows from Hermitian Operator is Normal and Eigenvalues of Normal Operator have Orthogonal Eigenspaces.
{{qed}} | The [[Definition:Eigenvector of Linear Operator|eigenvectors]] of a [[Definition:Hermitian Operator|Hermitian operator]] are [[Definition:Orthogonal (Linear Algebra)|orthogonal]]. | Follows from [[Hermitian Operator is Normal]] and [[Eigenvalues of Normal Operator have Orthogonal Eigenspaces]].
{{qed}} | Hermitian Operator has Orthogonal Eigenvectors/Proof 2 | https://proofwiki.org/wiki/Hermitian_Operator_has_Orthogonal_Eigenvectors | https://proofwiki.org/wiki/Hermitian_Operator_has_Orthogonal_Eigenvectors/Proof_2 | [
"Hermitian Operators",
"Eigenvectors of Linear Operators",
"Hermitian Operator has Orthogonal Eigenvectors"
] | [
"Definition:Eigenvector/Linear Operator",
"Definition:Hermitian Operator",
"Definition:Orthogonal (Linear Algebra)"
] | [
"Hermitian Operator is Normal",
"Eigenvalues of Normal Operator have Orthogonal Eigenspaces"
] |
proofwiki-4061 | Hausdorff Space iff Diagonal Set on Product is Closed | Let $T = \struct {S, \tau}$ be a topological space.
Let $\Delta_S$ be the diagonal set on $S$:
:$\Delta_S = \set {\tuple {x, x} \in S \times S: x \in S}$
where $S \times S$ is the Cartesian product of $S$ with itself.
Let $T^2 = \struct {S \times S, \TT}$ be the product space with product topology $\TT$.
Then $T$ is a ... | Suppose $T$ is a $T_2$ (Hausdorff) space.
Let $\Delta: S \to S \times S$ be the diagonal mapping on $S$.
We have that $\map \Delta S$ consists of all the elements of $S \times S$ whose coordinates are equal.
:$\map \Delta S = \set {\tuple {x, x} \in S \times S: x \in S} = \set {\tuple {x, y} \in S \times S: x = y}$
By ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\Delta_S$ be the [[Definition:Diagonal Relation|diagonal set]] on $S$:
:$\Delta_S = \set {\tuple {x, x} \in S \times S: x \in S}$
where $S \times S$ is the [[Definition:Cartesian Product|Cartesian product]] of $S$ with itself.
L... | Suppose $T$ is a [[Definition:Hausdorff Space|$T_2$ (Hausdorff) space]].
Let $\Delta: S \to S \times S$ be the [[Definition:Diagonal Mapping|diagonal mapping]] on $S$.
We have that $\map \Delta S$ consists of all the elements of $S \times S$ whose coordinates are equal.
:$\map \Delta S = \set {\tuple {x, x} \in S \ti... | Hausdorff Space iff Diagonal Set on Product is Closed | https://proofwiki.org/wiki/Hausdorff_Space_iff_Diagonal_Set_on_Product_is_Closed | https://proofwiki.org/wiki/Hausdorff_Space_iff_Diagonal_Set_on_Product_is_Closed | [
"Hausdorff Spaces",
"Product Topology"
] | [
"Definition:Topological Space",
"Definition:Diagonal Relation",
"Definition:Cartesian Product",
"Definition:Product Space (Topology)",
"Definition:Product Topology",
"Definition:T2 Space",
"Definition:Closed Set/Topology"
] | [
"Definition:T2 Space",
"Definition:Diagonal Mapping",
"Definition:Relative Complement",
"Definition:T2 Space",
"Natural Basis of Product Topology/Finite Product",
"Definition:Product Topology/Natural Basis",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Top... |
proofwiki-4062 | T0 Property is Hereditary | :The property of being a $T_0$ space is hereditary. | Let $T$ be a a $T_0$ space.
{{Recall|T0 Space|$T_0$ space}}
{{:Definition:T0 Space/Definition 1}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $x, y \in H$ such that $x \ne y$.
Then as $x, y \in S$ we have that:
:$\exists U \in \tau: x \in U, y \notin U$
or:
:$\exists U \in ... | :The property of being a [[Definition:T0 Space|$T_0$ space]] is [[Definition:Hereditary Property (Topology)|hereditary]]. | Let $T$ be a a [[Definition:T0 Space|$T_0$ space]].
{{Recall|T0 Space|$T_0$ space}}
{{:Definition:T0 Space/Definition 1}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $x, y \in H$ such that $x \ne y$.
Then as $x, y \in S$ we have that:
:$\exists U \in \tau: x \in U, y \... | T0 Property is Hereditary | https://proofwiki.org/wiki/T0_Property_is_Hereditary | https://proofwiki.org/wiki/T0_Property_is_Hereditary | [
"T0 Spaces",
"Separation Properties Preserved in Subspace",
"Examples of Hereditary Properties"
] | [
"Definition:T0 Space",
"Definition:Hereditary Property (Topology)"
] | [
"Definition:T0 Space",
"Definition:T0 Space"
] |
proofwiki-4063 | T1 Property is Hereditary | :The property of being a $T_1$ space is hereditary. | Let $T$ be a $T_1$ space.
{{Recall|T1 Space|$T_1$ space}}
{{:Definition:T1 Space/Definition 1}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $x \in H$ such that $x \ne y$.
Then as $x \in S$:
:$\forall y \in S: \exists U \in \tau: x \in U, y \notin U$
Then:
:$U \cap H \in \ta... | :The property of being a [[Definition:T1 Space|$T_1$ space]] is [[Definition:Hereditary Property (Topology)|hereditary]]. | Let $T$ be a [[Definition:T1 Space|$T_1$ space]].
{{Recall|T1 Space|$T_1$ space}}
{{:Definition:T1 Space/Definition 1}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $x \in H$ such that $x \ne y$.
Then as $x \in S$:
:$\forall y \in S: \exists U \in \tau: x \in U, y \noti... | T1 Property is Hereditary | https://proofwiki.org/wiki/T1_Property_is_Hereditary | https://proofwiki.org/wiki/T1_Property_is_Hereditary | [
"T1 Spaces",
"Separation Properties Preserved in Subspace",
"Examples of Hereditary Properties"
] | [
"Definition:T1 Space",
"Definition:Hereditary Property (Topology)"
] | [
"Definition:T1 Space",
"Definition:T1 Space"
] |
proofwiki-4064 | T2 Property is Hereditary | :The property of being a $T_2$ (Hausdorff) space is hereditary. | Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space.
{{Recall|T2 Space|$T_2$ (Hausdorff) space}}
{{:Definition:T2 Space/Definition 1}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $x, y \in H$ such that $x \ne y$.
Then as $x, y \in S$ we have that:
:$\exists U, V \in \t... | :The property of being a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] is [[Definition:Hereditary Property (Topology)|hereditary]]. | Let $T = \struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
{{Recall|T2 Space|$T_2$ (Hausdorff) space}}
{{:Definition:T2 Space/Definition 1}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $x, y \in H$ such that $x \ne y$.
Then as $x, y \in S$ we hav... | T2 Property is Hereditary | https://proofwiki.org/wiki/T2_Property_is_Hereditary | https://proofwiki.org/wiki/T2_Property_is_Hereditary | [
"Hausdorff Spaces",
"Separation Properties Preserved in Subspace",
"Examples of Hereditary Properties"
] | [
"Definition:T2 Space",
"Definition:Hereditary Property (Topology)"
] | [
"Definition:T2 Space",
"Definition:T2 Space"
] |
proofwiki-4065 | T2.5 Property is Hereditary | :The property of being a $T_{2 \frac 1 2}$ space is hereditary. | Let $T = \struct {S, \tau}$ be a $T_{2 \frac 1 2}$ space.
{{Recall|T2.5 Space|$T_{2 \frac 1 2}$ space}}
{{:Definition:T2.5 Space/Definition 1}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $x, y \in H$ such that $x \ne y$.
Then as $x, y \in S$ we have that:
:$\exists U, V \i... | :The property of being a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]] is [[Definition:Hereditary Property (Topology)|hereditary]]. | Let $T = \struct {S, \tau}$ be a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]].
{{Recall|T2.5 Space|$T_{2 \frac 1 2}$ space}}
{{:Definition:T2.5 Space/Definition 1}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $x, y \in H$ such that $x \ne y$.
Then as $x, y \in S$ ... | T2.5 Property is Hereditary | https://proofwiki.org/wiki/T2.5_Property_is_Hereditary | https://proofwiki.org/wiki/T2.5_Property_is_Hereditary | [
"T2.5 Spaces",
"Separation Properties Preserved in Subspace",
"Examples of Hereditary Properties"
] | [
"Definition:T2.5 Space",
"Definition:Hereditary Property (Topology)"
] | [
"Definition:T2.5 Space",
"Definition:Closure (Topology)",
"Definition:Set",
"Definition:T2.5 Space"
] |
proofwiki-4066 | T3 Property is Hereditary | :The property of being a $T_3$ space is hereditary. | Let $T = \struct {S, \tau}$ be a $T_3$ space.
{{Recall|T3 Space|$T_3$ space}}
{{:Definition:T3 Space/Definition 1}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $F \subseteq H$ such that $F$ is closed in $H$.
Let $y \in H$ such that $y \notin F$.
From Closed Set in Topologic... | :The property of being a [[Definition:T3 Space|$T_3$ space]] is [[Definition:Hereditary Property (Topology)|hereditary]]. | Let $T = \struct {S, \tau}$ be a [[Definition:T3 Space|$T_3$ space]].
{{Recall|T3 Space|$T_3$ space}}
{{:Definition:T3 Space/Definition 1}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $F \subseteq H$ such that $F$ is [[Definition:Closed Set (Topology)|closed]] in $H$.
... | T3 Property is Hereditary | https://proofwiki.org/wiki/T3_Property_is_Hereditary | https://proofwiki.org/wiki/T3_Property_is_Hereditary | [
"T3 Spaces",
"Separation Properties Preserved in Subspace",
"Examples of Hereditary Properties"
] | [
"Definition:T3 Space",
"Definition:Hereditary Property (Topology)"
] | [
"Definition:T3 Space",
"Definition:Closed Set/Topology",
"Closed Set in Topological Subspace",
"Definition:Closed Set/Topology",
"Definition:T3 Space",
"Definition:T3 Space",
"Category:T3 Spaces",
"Category:Separation Properties Preserved in Subspace",
"Category:Examples of Hereditary Properties"
] |
proofwiki-4067 | T3.5 Property is Hereditary | :The property of being a $T_{3 \frac 1 2}$ space is hereditary. | Let $T = \struct {S, \tau}$ be a $T_{3 \frac 1 2}$ space.
{{Recall|T3.5 Space|$T_{3 \frac 1 2}$ space}}
{{:Definition:T3.5 Space}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $F \subseteq H$ such that $F$ is closed in $H$.
Let $y \in H$ such that $y \notin F$.
From Closed S... | :The property of being a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]] is [[Definition:Hereditary Property (Topology)|hereditary]]. | Let $T = \struct {S, \tau}$ be a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]].
{{Recall|T3.5 Space|$T_{3 \frac 1 2}$ space}}
{{:Definition:T3.5 Space}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $F \subseteq H$ such that $F$ is [[Definition:Closed Set (Topology)|c... | T3.5 Property is Hereditary | https://proofwiki.org/wiki/T3.5_Property_is_Hereditary | https://proofwiki.org/wiki/T3.5_Property_is_Hereditary | [
"T3.5 Spaces",
"Separation Properties Preserved in Subspace",
"Examples of Hereditary Properties"
] | [
"Definition:T3.5 Space",
"Definition:Hereditary Property (Topology)"
] | [
"Definition:T3.5 Space",
"Definition:Closed Set/Topology",
"Closed Set in Topological Subspace",
"Definition:Closed Set/Topology",
"Definition:T3.5 Space",
"Definition:Urysohn Function",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Real Interval/Unit Interval/Closed",
"Definiti... |
proofwiki-4068 | T5 Property is Hereditary | :The property of being a $T_5$ space is hereditary. | Let $T = \struct {S, \tau}$ be a $T_5$ space.
{{Recall|T5 Space|$T_5$ space}}
{{:Definition:T5 Space/Definition 1}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $A, B \subseteq H$ such that $\map {\cl_H} A \cap B = A \cap \map {\cl_H} B = \O$.
That is, $A$ and $B$ are separa... | :The property of being a [[Definition:T5 Space|$T_5$ space]] is [[Definition:Hereditary Property (Topology)|hereditary]]. | Let $T = \struct {S, \tau}$ be a [[Definition:T5 Space|$T_5$ space]].
{{Recall|T5 Space|$T_5$ space}}
{{:Definition:T5 Space/Definition 1}}
We have that the set $\tau_H$ is defined as:
:$\tau_H := \set {U \cap H: U \in \tau}$
Let $A, B \subseteq H$ such that $\map {\cl_H} A \cap B = A \cap \map {\cl_H} B = \O$.
Th... | T5 Property is Hereditary | https://proofwiki.org/wiki/T5_Property_is_Hereditary | https://proofwiki.org/wiki/T5_Property_is_Hereditary | [
"T5 Spaces",
"Separation Properties Preserved in Subspace",
"Examples of Hereditary Properties"
] | [
"Definition:T5 Space",
"Definition:Hereditary Property (Topology)"
] | [
"Definition:T5 Space",
"Definition:Separated Sets",
"Intersection is Associative",
"Closure of Subset in Subspace",
"Definition:T5 Space",
"Definition:T5 Space",
"Category:T5 Spaces",
"Category:Separation Properties Preserved in Subspace",
"Category:Examples of Hereditary Properties"
] |
proofwiki-4069 | Product Space is T1 iff Factor Spaces are T1 | :$T$ is a $T_1$ space {{iff}} each of $\struct{S_\alpha, \tau_\alpha}$ is a $T_1$ space. | === Sufficient Condition ===
{{:Product Space is T1 iff Factor Spaces are T1/Sufficient Condition/Proof 1}}{{qed|lemma}} | :$T$ is a [[Definition:T1 Space|$T_1$ space]] {{iff}} each of $\struct{S_\alpha, \tau_\alpha}$ is a [[Definition:T1 Space|$T_1$ space]]. | === [[Product Space is T1 iff Factor Spaces are T1/Sufficient Condition|Sufficient Condition]] ===
{{:Product Space is T1 iff Factor Spaces are T1/Sufficient Condition/Proof 1}}{{qed|lemma}} | Product Space is T1 iff Factor Spaces are T1 | https://proofwiki.org/wiki/Product_Space_is_T1_iff_Factor_Spaces_are_T1 | https://proofwiki.org/wiki/Product_Space_is_T1_iff_Factor_Spaces_are_T1 | [
"Product Space is T1 iff Factor Spaces are T1",
"T1 Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:T1 Space",
"Definition:T1 Space"
] | [
"Product Space is T1 iff Factor Spaces are T1/Sufficient Condition"
] |
proofwiki-4070 | Product Space is T1 iff Factor Spaces are T1 | :$T$ is a $T_1$ space {{iff}} each of $\struct{S_\alpha, \tau_\alpha}$ is a $T_1$ space. | Let $T$ be a $T_1$ space.
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the axiom of choice.
From Subspace of Product Space is Homeomorphic to Factor Space, $\struct {S_\alpha, \tau_\alpha}$ is homeomorphic to a subspace $T_\alpha$ of $T$.
From $T_1$ property is hereditary, $T_\alpha$ is a $T_1$ space.
From $T_1$ Pro... | :$T$ is a [[Definition:T1 Space|$T_1$ space]] {{iff}} each of $\struct{S_\alpha, \tau_\alpha}$ is a [[Definition:T1 Space|$T_1$ space]]. | Let $T$ be a [[Definition:T1 Space|$T_1$ space]].
As $S_\alpha \ne \O$ we also have $S \ne \O$ by the [[Axiom:Axiom of Choice|axiom of choice]].
From [[Subspace of Product Space is Homeomorphic to Factor Space]], $\struct {S_\alpha, \tau_\alpha}$ is [[Definition:Homeomorphism (Topological Spaces)|homeomorphic]] to a ... | Product Space is T1 iff Factor Spaces are T1/Sufficient Condition/Proof 1 | https://proofwiki.org/wiki/Product_Space_is_T1_iff_Factor_Spaces_are_T1 | https://proofwiki.org/wiki/Product_Space_is_T1_iff_Factor_Spaces_are_T1/Sufficient_Condition/Proof_1 | [
"Product Space is T1 iff Factor Spaces are T1",
"T1 Spaces",
"Separation Properties Preserved under Topological Product",
"Product Spaces"
] | [
"Definition:T1 Space",
"Definition:T1 Space"
] | [
"Definition:T1 Space",
"Axiom:Axiom of Choice",
"Subspace of Product Space is Homeomorphic to Factor Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Subspace",
"T1 Property is Hereditary",
"Definition:T1 Space",
"T1 Property is Preserved under Homeomorphism",
"Definition:T1 Space"... |
proofwiki-4071 | Gödel's Incompleteness Theorems/First | Let $T$ be the set of theorems of some recursive set of sentences in the language of arithmetic such that $T$ contains minimal arithmetic.
$T$ cannot be both consistent and complete. | {{AimForCont}} that such a $T$ is consistent and complete.
By the Undecidability Theorem, since $T$ is consistent and contains $Q$, it is not recursive.
But, by Complete Recursively Axiomatized Theories are Recursive, since $T$ is complete and is the set of theorems of a recursive set, it is recursive.
The result follo... | Let $T$ be the [[Definition:Set|set]] of [[Definition:Theorem of Logic|theorems]] of some [[Definition:Recursive Set|recursive set]] of [[Definition:Sentence|sentences]] in the [[Definition:Language of Arithmetic|language of arithmetic]] such that $T$ contains [[Definition:Minimal Arithmetic|minimal arithmetic]].
$T$... | {{AimForCont}} that such a $T$ is [[Definition:Consistent (Logic)|consistent]] and [[Definition:Complete Theory|complete]].
By the [[Undecidability Theorem]], since $T$ is [[Definition:Consistent (Logic)|consistent]] and contains $Q$, it is not [[Definition:Recursive Set|recursive]].
But, by [[Complete Recursively Ax... | Gödel's Incompleteness Theorems/First | https://proofwiki.org/wiki/Gödel's_Incompleteness_Theorems/First | https://proofwiki.org/wiki/Gödel's_Incompleteness_Theorems/First | [
"Gödel's Incompleteness Theorems"
] | [
"Definition:Set",
"Definition:Theorem/Logic",
"Definition:Recursive/Set",
"Definition:Classes of WFFs/Sentence",
"Definition:Language of Arithmetic",
"Definition:Minimal Arithmetic",
"Definition:Consistent (Logic)",
"Definition:Complete Theory"
] | [
"Definition:Consistent (Logic)",
"Definition:Complete Theory",
"Undecidability Theorem",
"Definition:Consistent (Logic)",
"Definition:Recursive/Set",
"Complete Recursively Axiomatized Theories are Recursive",
"Definition:Complete Theory",
"Definition:Recursive/Set",
"Definition:Recursive/Set",
"Pr... |
proofwiki-4072 | Undecidability Theorem | Let $T$ be the set of theorems of some consistent theory in the language of arithmetic which contains minimal arithmetic $Q$.
Then $T$ is not recursive. | Let $\Theta$ be the set of Gödel numbers of the theorems in $T$.
We have that $T$ is a consistent extension of $Q$.
By Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic, $\Theta$ is not a definable set in $T$.
Since Recursive Sets are Definable in Arithmetic, this means that $\Theta$ is not recurs... | Let $T$ be the set of [[Definition:Theorem of Logic|theorems]] of some [[Definition:Consistent (Logic)|consistent theory]] in the [[Definition:Language of Arithmetic|language of arithmetic]] which contains [[Definition:Minimal Arithmetic|minimal arithmetic]] $Q$.
Then $T$ is not [[Definition:Recursive Set|recursive]]... | Let $\Theta$ be the set of [[Definition:Gödel Number|Gödel numbers]] of the theorems in $T$.
We have that $T$ is a [[Definition:Consistent (Logic)|consistent]] extension of $Q$.
By [[Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic]], $\Theta$ is not a [[Definition:Definable Set|definable set]]... | Undecidability Theorem | https://proofwiki.org/wiki/Undecidability_Theorem | https://proofwiki.org/wiki/Undecidability_Theorem | [
"Mathematical Logic"
] | [
"Definition:Theorem/Logic",
"Definition:Consistent (Logic)",
"Definition:Language of Arithmetic",
"Definition:Minimal Arithmetic",
"Definition:Recursive/Set"
] | [
"Definition:Gödel Number",
"Definition:Consistent (Logic)",
"Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic",
"Definition:Definable/Set",
"Recursive Sets are Definable in Arithmetic",
"Definition:Recursive/Set",
"Definition:Recursive/Set",
"Gödel Numbering is Recursive",
"De... |
proofwiki-4073 | Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic | Let $T$ be the set of theorems of some consistent theory in the language of arithmetic which contains minimal arithmetic.
The set of Gödel numbers of the theorems of $T$ is not definable in $T$. | {{Questionable|Not only the proof is faulty, this theorem is wrong. If we have a recursively enumerable set A of axioms, then the set of theorems (and hence, the Gödel numbers of those) proven by A is recursively enumerable. Minimal arithmetic and PA are both recursively enumerable, and hence their theorems have a recu... | Let $T$ be the set of [[Definition:Theorem of Logic|theorems]] of some [[Definition:Consistent (Logic)|consistent theory]] in the [[Definition:Language of Arithmetic|language of arithmetic]] which contains [[Definition:Minimal Arithmetic|minimal arithmetic]].
The set of [[Definition:Gödel Number|Gödel numbers]] of th... | {{Questionable|Not only the proof is faulty, this theorem is wrong. If we have a recursively enumerable set A of axioms, then the set of theorems (and hence, the Gödel numbers of those) proven by A is recursively enumerable. Minimal arithmetic and PA are both recursively enumerable, and hence their theorems have a recu... | Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic | https://proofwiki.org/wiki/Set_of_Gödel_Numbers_of_Arithmetic_Theorems_Not_Definable_in_Arithmetic | https://proofwiki.org/wiki/Set_of_Gödel_Numbers_of_Arithmetic_Theorems_Not_Definable_in_Arithmetic | [
"Mathematical Logic"
] | [
"Definition:Theorem/Logic",
"Definition:Consistent (Logic)",
"Definition:Language of Arithmetic",
"Definition:Minimal Arithmetic",
"Definition:Gödel Number",
"Definition:Definable"
] | [
"Definition:Gödel Number",
"Diagonal Lemma"
] |
proofwiki-4074 | Diagonal Lemma | Let $T$ be the set of theorems of some theory in the language of arithmetic which contains minimal arithmetic.
For any formula $\map B y$ in the language of arithmetic, there is a sentence $G$ such that
:$T \vdash G \leftrightarrow \map B {\hat G}$
where $\hat G$ is the Gödel number of $G$ (more accurately, it is the t... | There is a primitive recursive function $\mathrm {diag}$ which is defined by:
:$\map {\mathrm {diag} } n = \widehat {\map A {\hat A} }$
where:
:$\map A x$ is the formula such that $\hat A = n$.
:the $\hat {}$ sign denotes the Gödel number of the contained formula (and we are not being formal about distinguishing betwee... | Let $T$ be the set of [[Definition:Theorem of Logic|theorems]] of some [[Definition:Theory|theory]] in the [[Definition:Language of Arithmetic|language of arithmetic]] which contains [[Definition:Minimal Arithmetic|minimal arithmetic]].
For any [[Definition:Logical Formula|formula]] $\map B y$ in the language of arit... | There is a primitive recursive function $\mathrm {diag}$ which is defined by:
:$\map {\mathrm {diag} } n = \widehat {\map A {\hat A} }$
where:
:$\map A x$ is the formula such that $\hat A = n$.
:the $\hat {}$ sign denotes the Gödel number of the contained formula (and we are not being formal about distinguishing betwe... | Diagonal Lemma | https://proofwiki.org/wiki/Diagonal_Lemma | https://proofwiki.org/wiki/Diagonal_Lemma | [
"Model Theory for Predicate Logic"
] | [
"Definition:Theorem/Logic",
"Definition:Theory",
"Definition:Language of Arithmetic",
"Definition:Minimal Arithmetic",
"Definition:Logical Formula"
] | [
"Recursive Sets are Definable in Arithmetic",
"Definition:Definable"
] |
proofwiki-4075 | Tarski's Undefinability Theorem | Let $\ZZ$ be the standard structure $\struct {\Z, +, \cdot, s, <, 0}$ for the language of arithmetic.
Let $\operatorname {Th}_\ZZ$ be the sentences which are true in $\ZZ$.
Let $\Theta$ be the set of Gödel numbers of those sentences in $\operatorname {Th}_\ZZ$.
$\Theta$ is not definable in $\operatorname {Th}_\ZZ$. | $\operatorname {Th}_\ZZ$ is easily seen to be a consistent extension of minimal arithmetic. (In fact, the axioms in minimal arithmetic were selected based on the behavior of standard arithmetic.)
Thus, the theorem is a special case of Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic (and can be ... | Let $\ZZ$ be the standard structure $\struct {\Z, +, \cdot, s, <, 0}$ for the [[Definition:Language of Arithmetic|language of arithmetic]].
Let $\operatorname {Th}_\ZZ$ be the [[Definition:Sentence|sentences]] which are [[Definition:Satisfiable|true]] in $\ZZ$.
Let $\Theta$ be the set of [[Definition:Gödel Number|Göd... | $\operatorname {Th}_\ZZ$ is easily seen to be a [[Definition:Consistent (Logic)|consistent]] extension of [[Definition:Minimal Arithmetic|minimal arithmetic]]. (In fact, the axioms in minimal arithmetic were selected based on the behavior of standard arithmetic.)
Thus, the theorem is a special case of [[Set of Gödel... | Tarski's Undefinability Theorem | https://proofwiki.org/wiki/Tarski's_Undefinability_Theorem | https://proofwiki.org/wiki/Tarski's_Undefinability_Theorem | [
"Mathematical Logic"
] | [
"Definition:Language of Arithmetic",
"Definition:Classes of WFFs/Sentence",
"Definition:Satisfiable",
"Definition:Gödel Number",
"Definition:Definable"
] | [
"Definition:Consistent (Logic)",
"Definition:Minimal Arithmetic",
"Set of Gödel Numbers of Arithmetic Theorems Not Definable in Arithmetic"
] |
proofwiki-4076 | Equivalence of Definitions of Limit Point of Set | Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$.
{{TFAE|def = Limit Point of Set|view = a limit point of a set}}
=== Definition from Open Neighborhood===
{{:Definition:Limit Point of Set/Definition from Open Neighborhood}}
=== Definition from Closure===
{{:Definition:Limit Point of Set/Definitio... | === Definition from Open Neighborhood $\iff$ Definition from Closure ===
The closure of $A$ is defined as the union of the set of all isolated points of $A$ and the set of all limit points of $A$.
The rest then follows directly from Equivalence of Definitions of Isolated Point.
{{qed|lemma}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A \subseteq S$.
{{TFAE|def = Limit Point of Set|view = a limit point of a set}}
=== [[Definition:Limit Point of Set/Definition from Open Neighborhood|Definition from Open Neighborhood]]===
{{:Definition:Limit Point of Set/Defi... | === Definition from Open Neighborhood $\iff$ Definition from Closure ===
The [[Definition:Closure (Topology)|closure]] of $A$ is defined as the [[Definition:Set Union|union]] of the set of all [[Definition:Isolated Point (Topology)|isolated points]] of $A$ and the set of all [[Definition:Limit Point of Set|limit point... | Equivalence of Definitions of Limit Point of Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Limit_Point_of_Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Limit_Point_of_Set | [
"Equivalence of Definitions of Limit Point of Set",
"Limit Points of Sets"
] | [
"Definition:Topological Space",
"Definition:Limit Point of Set/Definition from Open Neighborhood",
"Definition:Limit Point of Set/Definition from Closure",
"Definition:Limit Point of Set/Definition from Adherent Point",
"Definition:Limit Point of Set/Definition from Relative Complement"
] | [
"Definition:Closure (Topology)",
"Definition:Set Union",
"Definition:Isolated Point (Topology)",
"Definition:Limit Point/Topology/Set",
"Equivalence of Definitions of Isolated Point"
] |
proofwiki-4077 | Equivalence of Definitions of Limit Point of Set | Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$.
{{TFAE|def = Limit Point of Set|view = a limit point of a set}}
=== Definition from Open Neighborhood===
{{:Definition:Limit Point of Set/Definition from Open Neighborhood}}
=== Definition from Closure===
{{:Definition:Limit Point of Set/Definitio... | The following equivalence holds:
{{begin-eqn}}
{{eqn | o =
| c = There exists an open neighborhood $U$ of $x$ such that $A \cap \paren {U \setminus \set x} = \O$
}}
{{eqn | o = \leadstoandfrom
| c = There exists an open neighborhood $U$ of $x$ such that $U \subseteq \paren{S \setminus A} \cup \set x$
... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A \subseteq S$.
{{TFAE|def = Limit Point of Set|view = a limit point of a set}}
=== [[Definition:Limit Point of Set/Definition from Open Neighborhood|Definition from Open Neighborhood]]===
{{:Definition:Limit Point of Set/Defi... | The following [[Definition:Logical Equivalence|equivalence]] holds:
{{begin-eqn}}
{{eqn | o =
| c = There exists an [[Definition:Open Neighborhood of Point|open neighborhood]] $U$ of $x$ such that $A \cap \paren {U \setminus \set x} = \O$
}}
{{eqn | o = \leadstoandfrom
| c = There exists an [[Definition:Ope... | Equivalence of Definitions of Limit Point of Set/Definition from Open Neighborhood iff Definition from Relative Complement/Proof 1 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Limit_Point_of_Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Limit_Point_of_Set/Definition_from_Open_Neighborhood_iff_Definition_from_Relative_Complement/Proof_1 | [
"Equivalence of Definitions of Limit Point of Set",
"Limit Points of Sets"
] | [
"Definition:Topological Space",
"Definition:Limit Point of Set/Definition from Open Neighborhood",
"Definition:Limit Point of Set/Definition from Closure",
"Definition:Limit Point of Set/Definition from Adherent Point",
"Definition:Limit Point of Set/Definition from Relative Complement"
] | [
"Definition:Logical Equivalence",
"Definition:Open Neighborhood/Point",
"Definition:Open Neighborhood/Point",
"Modus Ponendo Tollens",
"Definition:Neighborhood (Topology)/Point",
"Rule of Transposition"
] |
proofwiki-4078 | Equivalence of Definitions of Limit Point of Set | Let $T = \struct {S, \tau}$ be a topological space.
Let $A \subseteq S$.
{{TFAE|def = Limit Point of Set|view = a limit point of a set}}
=== Definition from Open Neighborhood===
{{:Definition:Limit Point of Set/Definition from Open Neighborhood}}
=== Definition from Closure===
{{:Definition:Limit Point of Set/Definitio... | The following equivalence holds:
There exists an open neighborhood $U$ of $x$ such that $A \cap \paren {U \setminus \set x} = \O$
{{begin-eqn}}
{{eqn | l = \O
| r = A \cap \paren {U \setminus \set x}
}}
{{eqn | ll = \leadstoandfrom
| l = \O
| r = \paren {U \cap A} \setminus \set x
| cc = Inter... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $A \subseteq S$.
{{TFAE|def = Limit Point of Set|view = a limit point of a set}}
=== [[Definition:Limit Point of Set/Definition from Open Neighborhood|Definition from Open Neighborhood]]===
{{:Definition:Limit Point of Set/Defi... | The following [[Definition:Logical Equivalence|equivalence]] holds:
There exists an [[Definition:Open Neighborhood of Point|open neighborhood]] $U$ of $x$ such that $A \cap \paren {U \setminus \set x} = \O$
{{begin-eqn}}
{{eqn | l = \O
| r = A \cap \paren {U \setminus \set x}
}}
{{eqn | ll = \leadstoandfrom
... | Equivalence of Definitions of Limit Point of Set/Definition from Open Neighborhood iff Definition from Relative Complement/Proof 2 | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Limit_Point_of_Set | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Limit_Point_of_Set/Definition_from_Open_Neighborhood_iff_Definition_from_Relative_Complement/Proof_2 | [
"Equivalence of Definitions of Limit Point of Set",
"Limit Points of Sets"
] | [
"Definition:Topological Space",
"Definition:Limit Point of Set/Definition from Open Neighborhood",
"Definition:Limit Point of Set/Definition from Closure",
"Definition:Limit Point of Set/Definition from Adherent Point",
"Definition:Limit Point of Set/Definition from Relative Complement"
] | [
"Definition:Logical Equivalence",
"Definition:Open Neighborhood/Point",
"Intersection with Set Difference is Set Difference with Intersection",
"Intersection is Commutative",
"Intersection with Set Difference is Set Difference with Intersection",
"Complement of Complement",
"Intersection with Complement... |
proofwiki-4079 | Denseness Preserved in Coarser Topology | Let $T = \struct {S, \tau}$ be a topological space.
Let $T' = \struct {S, \tau'}$ be another topological space on $S$ such that $\tau'$ is coarser than $\tau$.
Let $H \subseteq S$ be everywhere dense in $T$.
Then $H$ is also everywhere dense in $T'$. | Let $H \subseteq S$ be everywhere dense in $T$.
As $\tau'$ is coarser than $\tau$, by definition $\tau' \subseteq \tau$.
Let $x \in S$.
As $H \subseteq S$ is everywhere dense in $T$, either $x \in H$ or $x$ is a limit point of $H$ in $T$.
Suppose $x \notin H$.
Then $x$ is a limit point of $H$ in $T$.
Then by definitio... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T' = \struct {S, \tau'}$ be another [[Definition:Topological Space|topological space]] on $S$ such that $\tau'$ is [[Definition:Coarser Topology|coarser]] than $\tau$.
Let $H \subseteq S$ be [[Definition:Everywhere Dense|everyw... | Let $H \subseteq S$ be [[Definition:Everywhere Dense|everywhere dense]] in $T$.
As $\tau'$ is [[Definition:Coarser Topology|coarser]] than $\tau$, by definition $\tau' \subseteq \tau$.
Let $x \in S$.
As $H \subseteq S$ is [[Definition:Everywhere Dense|everywhere dense]] in $T$, either $x \in H$ or $x$ is a [[Defini... | Denseness Preserved in Coarser Topology | https://proofwiki.org/wiki/Denseness_Preserved_in_Coarser_Topology | https://proofwiki.org/wiki/Denseness_Preserved_in_Coarser_Topology | [
"Denseness"
] | [
"Definition:Topological Space",
"Definition:Topological Space",
"Definition:Coarser Topology",
"Definition:Everywhere Dense",
"Definition:Everywhere Dense"
] | [
"Definition:Everywhere Dense",
"Definition:Coarser Topology",
"Definition:Everywhere Dense",
"Definition:Limit Point/Topology/Set",
"Definition:Limit Point/Topology/Set",
"Definition:Limit Point/Topology/Set",
"Definition:Limit Point/Topology/Set",
"Definition:Limit Point/Topology/Set",
"Definition:... |
proofwiki-4080 | Rationals are Everywhere Dense in Compact Complement Space | Let $T = \struct {\R, \tau^*}$ be the compact complement space on $\R$.
Let $\Q$ be the set of rational numbers.
Then $\Q$ is everywhere dense in $T$. | We have that the Compact Complement Topology is Coarser than Euclidean Topology.
The result follows from Denseness Preserved in Coarser Topology.
{{qed}} | Let $T = \struct {\R, \tau^*}$ be the [[Definition:Compact Complement Space|compact complement space]] on $\R$.
Let $\Q$ be the [[Definition:Rational Number|set of rational numbers]].
Then $\Q$ is [[Definition:Everywhere Dense|everywhere dense]] in $T$. | We have that the [[Compact Complement Topology is Coarser than Euclidean Topology]].
The result follows from [[Denseness Preserved in Coarser Topology]].
{{qed}} | Rationals are Everywhere Dense in Compact Complement Space | https://proofwiki.org/wiki/Rationals_are_Everywhere_Dense_in_Compact_Complement_Space | https://proofwiki.org/wiki/Rationals_are_Everywhere_Dense_in_Compact_Complement_Space | [
"Rational Number Space",
"Compact Complement Topology",
"Examples of Everywhere Dense"
] | [
"Definition:Compact Complement Topology",
"Definition:Rational Number",
"Definition:Everywhere Dense"
] | [
"Compact Complement Topology is Coarser than Euclidean Topology",
"Denseness Preserved in Coarser Topology"
] |
proofwiki-4081 | Compact Complement Topology is Separable | Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$.
Then $T$ is a separable space. | We have:
: Compact Complement Topology is Second-Countable
: Second-Countable Space is Separable
Hence the result.
{{qed}} | Let $T = \struct {\R, \tau}$ be the [[Definition:Compact Complement Topology|compact complement topology]] on $\R$.
Then $T$ is a [[Definition:Separable Space|separable space]]. | We have:
: [[Compact Complement Topology is Second-Countable]]
: [[Second-Countable Space is Separable]]
Hence the result.
{{qed}} | Compact Complement Topology is Separable/Proof 1 | https://proofwiki.org/wiki/Compact_Complement_Topology_is_Separable | https://proofwiki.org/wiki/Compact_Complement_Topology_is_Separable/Proof_1 | [
"Compact Complement Topology is Separable",
"Compact Complement Topology",
"Examples of Separable Spaces"
] | [
"Definition:Compact Complement Topology",
"Definition:Separable Space"
] | [
"Compact Complement Topology is Second-Countable",
"Second-Countable Space is Separable"
] |
proofwiki-4082 | Compact Complement Topology is Separable | Let $T = \struct {\R, \tau}$ be the compact complement topology on $\R$.
Then $T$ is a separable space. | {{Recall|Separable Space|separable space}}
{{:Definition:Separable Space}}
We have that:
:Rationals are Everywhere Dense in Compact Complement Space
:Rational Numbers are Countably Infinite
Hence the result by definition of separable space.
{{qed}} | Let $T = \struct {\R, \tau}$ be the [[Definition:Compact Complement Topology|compact complement topology]] on $\R$.
Then $T$ is a [[Definition:Separable Space|separable space]]. | {{Recall|Separable Space|separable space}}
{{:Definition:Separable Space}}
We have that:
:[[Rationals are Everywhere Dense in Compact Complement Space]]
:[[Rational Numbers are Countably Infinite]]
Hence the result by definition of [[Definition:Separable Space|separable space]].
{{qed}} | Compact Complement Topology is Separable/Proof 2 | https://proofwiki.org/wiki/Compact_Complement_Topology_is_Separable | https://proofwiki.org/wiki/Compact_Complement_Topology_is_Separable/Proof_2 | [
"Compact Complement Topology is Separable",
"Compact Complement Topology",
"Examples of Separable Spaces"
] | [
"Definition:Compact Complement Topology",
"Definition:Separable Space"
] | [
"Rationals are Everywhere Dense in Compact Complement Space",
"Rational Numbers are Countably Infinite",
"Definition:Separable Space"
] |
proofwiki-4083 | Either-Or Topology is Locally Connected | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is a locally connected space. | :Either-Or Topology is Locally Path-Connected
:Locally Path-Connected Space is Locally Connected
{{qed}} | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Then $T$ is a [[Definition:Locally Connected Space|locally connected space]]. | :[[Either-Or Topology is Locally Path-Connected]]
:[[Locally Path-Connected Space is Locally Connected]]
{{qed}} | Either-Or Topology is Locally Connected | https://proofwiki.org/wiki/Either-Or_Topology_is_Locally_Connected | https://proofwiki.org/wiki/Either-Or_Topology_is_Locally_Connected | [
"Either-Or Topology",
"Examples of Locally Connected Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:Locally Connected Space"
] | [
"Either-Or Topology is Locally Path-Connected",
"Locally Path-Connected Space is Locally Connected"
] |
proofwiki-4084 | Either-Or Topology is not Locally Injectively Path-Connected | Let $T = \struct {S, \tau}$ be the either-or space.
Then $T$ is not a locally injectively path-connected space. | Let $\BB$ be a basis for $\tau$.
Suppose that $0 \in B$, where $B \in \BB$.
Then by definition of the either-or topology, $\openint {-1} 1 \subseteq B$.
In particular, $\dfrac 1 2 \in B$.
Let $f: \closedint 0 1 \to B$ be an arbitrary injection such that:
{{begin-eqn}}
{{eqn | l = \map f 0
| r = 0
}}
{{eqn | l = \... | Let $T = \struct {S, \tau}$ be the [[Definition:Either-Or Topology|either-or space]].
Then $T$ is not a [[Definition:Locally Injectively Path-Connected Space|locally injectively path-connected space]]. | Let $\BB$ be a [[Definition:Basis (Topology)|basis]] for $\tau$.
Suppose that $0 \in B$, where $B \in \BB$.
Then by definition of the [[Definition:Either-Or Topology|either-or topology]], $\openint {-1} 1 \subseteq B$.
In particular, $\dfrac 1 2 \in B$.
Let $f: \closedint 0 1 \to B$ be an [[Definition:Arbitrary|ar... | Either-Or Topology is not Locally Injectively Path-Connected | https://proofwiki.org/wiki/Either-Or_Topology_is_not_Locally_Injectively_Path-Connected | https://proofwiki.org/wiki/Either-Or_Topology_is_not_Locally_Injectively_Path-Connected | [
"Either-Or Topology",
"Examples of Locally Injectively Path-Connected Spaces"
] | [
"Definition:Either-Or Topology",
"Definition:Locally Injectively Path-Connected Space"
] | [
"Definition:Basis (Topology)",
"Definition:Either-Or Topology",
"Definition:Arbitrary",
"Definition:Injection",
"Definition:Open Set/Topology",
"Definition:Injection",
"Closed Real Interval is not Open Set",
"Definition:Open Set/Topology",
"Definition:Continuous Mapping (Topology)/Everywhere",
"De... |
proofwiki-4085 | Limit Points in Uncountable Fort Space | Let $T = \struct {S, \tau_p}$ be an uncountable Fort space.
Let $U \subseteq S$ be a countably infinite subset of $S$.
Then $p$ is the only limit point of $U$. | Suppose $y \in S, y \ne p$.
We have by definition of Fort space that $\set y$ is open in $T$.
So there is no $z \in \set y: z \ne y, z \in U$.
Hence $y$ can not be a limit point of $U$.
Suppose $p \in V \in \tau_p$ for some $V \subseteq S$.
From {{Defof|Fort Space}}, $\relcomp S V$ is finite.
Then:
{{begin-eqn}}
{{eqn... | Let $T = \struct {S, \tau_p}$ be an [[Definition:Uncountable Fort Space|uncountable Fort space]].
Let $U \subseteq S$ be a [[Definition:Countably Infinite Set|countably infinite]] [[Definition:Subset|subset]] of $S$.
Then $p$ is the only [[Definition:Limit Point of Set|limit point]] of $U$. | Suppose $y \in S, y \ne p$.
We have by definition of [[Definition:Fort Space|Fort space]] that $\set y$ is [[Definition:Open Set (Topology)|open]] in $T$.
So there is no $z \in \set y: z \ne y, z \in U$.
Hence $y$ can not be a [[Definition:Limit Point of Set|limit point of $U$]].
Suppose $p \in V \in \tau_p$ for ... | Limit Points in Uncountable Fort Space | https://proofwiki.org/wiki/Limit_Points_in_Uncountable_Fort_Space | https://proofwiki.org/wiki/Limit_Points_in_Uncountable_Fort_Space | [
"Uncountable Fort Spaces",
"Examples of Limit Points"
] | [
"Definition:Fort Space/Uncountable",
"Definition:Countably Infinite/Set",
"Definition:Subset",
"Definition:Limit Point/Topology/Set"
] | [
"Definition:Fort Space",
"Definition:Open Set/Topology",
"Definition:Limit Point/Topology/Set",
"Definition:Finite Set",
"Set Difference with Set Difference is Union of Set Difference with Intersection",
"Set Union Preserves Subsets",
"Intersection is Subset",
"Definition:Finite Set",
"Definition:Co... |
proofwiki-4086 | Fort Space is Scattered | Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$.
Then $T$ is a scattered space. | Let $H \subseteq T$ such that $H \ne \O$ and $H \ne \set p$.
Then $\exists x \in H: x \ne p$.
From Clopen Points in Fort Space, every point of $T$ apart from $p$ is open in $T$.
So $\set x$ is an open set of $T$.
So $H \cap \set x = \set x$ and so $x$ is isolated in $H$.
Thus $H$ contains at least one point which is is... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fort Space|Fort space]] on an [[Definition:Infinite Set|infinite set]] $S$.
Then $T$ is a [[Definition:Scattered Space|scattered space]]. | Let $H \subseteq T$ such that $H \ne \O$ and $H \ne \set p$.
Then $\exists x \in H: x \ne p$.
From [[Clopen Points in Fort Space]], every point of $T$ apart from $p$ is [[Definition:Open Point|open]] in $T$.
So $\set x$ is an [[Definition:Open Set (Topology)|open set]] of $T$.
So $H \cap \set x = \set x$ and so $x$... | Fort Space is Scattered/Proof 1 | https://proofwiki.org/wiki/Fort_Space_is_Scattered | https://proofwiki.org/wiki/Fort_Space_is_Scattered/Proof_1 | [
"Fort Space is Scattered",
"Fort Spaces",
"Examples of Scattered Spaces"
] | [
"Definition:Fort Space",
"Definition:Infinite Set",
"Definition:Scattered Space"
] | [
"Clopen Points in Fort Space",
"Definition:Open Point",
"Definition:Open Set/Topology",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Isolated Point (Topology)/Subset",
"Singleton Point is Isolated",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Isolated Point (Topology)/Subs... |
proofwiki-4087 | Fort Space is Scattered | Let $T = \struct {S, \tau_p}$ be a Fort space on an infinite set $S$.
Then $T$ is a scattered space. | {{Recall|Scattered Space|scattered space}}
{{:Definition:Scattered Space/Definition 1}}
{{AimForCont}} $H \subseteq T$ has no isolated points of $H$.
So, by definition, $H$ is dense in itself.
We have that:
:a Fort Space is $T_1$
:a Dense-in-itself Subset of $T_1$ Space is Infinite.
So $H$ is infinite, and so contains ... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Fort Space|Fort space]] on an [[Definition:Infinite Set|infinite set]] $S$.
Then $T$ is a [[Definition:Scattered Space|scattered space]]. | {{Recall|Scattered Space|scattered space}}
{{:Definition:Scattered Space/Definition 1}}
{{AimForCont}} $H \subseteq T$ has no [[Definition:Isolated Point of Subset|isolated points of $H$]].
So, by definition, $H$ is [[Definition:Dense-in-itself|dense in itself]].
We have that:
:a [[Fort Space is T1|Fort Space is $T_... | Fort Space is Scattered/Proof 2 | https://proofwiki.org/wiki/Fort_Space_is_Scattered | https://proofwiki.org/wiki/Fort_Space_is_Scattered/Proof_2 | [
"Fort Space is Scattered",
"Fort Spaces",
"Examples of Scattered Spaces"
] | [
"Definition:Fort Space",
"Definition:Infinite Set",
"Definition:Scattered Space"
] | [
"Definition:Isolated Point (Topology)/Subset",
"Definition:Dense-in-itself",
"Fort Space is T1",
"Dense-in-itself Subset of T1 Space is Infinite",
"Definition:Infinite Set",
"Definition:Element",
"Clopen Points in Fort Space",
"Definition:Open Set/Topology",
"Definition:Isolated Point (Topology)/Sub... |
proofwiki-4088 | Dense-in-itself Subset of T1 Space is Infinite | Let $T = \struct {S, \tau_p}$ be a $T_1$ space.
Let $H \subseteq T$ be dense-in-itself.
Then $H$ is infinite. | ;Proof by Contradiction
{{AimForCont}} $H$ is finite.
From Finite $T_1$ Space is Discrete, $H$ has the discrete topology.
From Discrete Space is not Dense-In-Itself it then follows that $H$ can not be dense-in-itself.
So for $H$ to be dense-in-itself, it must be infinite.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:T1 Space|$T_1$ space]].
Let $H \subseteq T$ be [[Definition:Dense-in-itself|dense-in-itself]].
Then $H$ is [[Definition:Infinite Set|infinite]]. | ;[[Proof by Contradiction]]
{{AimForCont}} $H$ is [[Definition:Finite Set|finite]].
From [[Finite T1 Space is Discrete|Finite $T_1$ Space is Discrete]], $H$ has the [[Definition:Discrete Topology|discrete topology]].
From [[Discrete Space is not Dense-In-Itself]] it then follows that $H$ can not be [[Definition:Dens... | Dense-in-itself Subset of T1 Space is Infinite | https://proofwiki.org/wiki/Dense-in-itself_Subset_of_T1_Space_is_Infinite | https://proofwiki.org/wiki/Dense-in-itself_Subset_of_T1_Space_is_Infinite | [
"Dense-in-itself",
"T1 Spaces",
"Infinite Sets"
] | [
"Definition:T1 Space",
"Definition:Dense-in-itself",
"Definition:Infinite Set"
] | [
"Proof by Contradiction",
"Definition:Finite Set",
"Finite T1 Space is Discrete",
"Definition:Discrete Topology",
"Discrete Space is not Dense-In-Itself",
"Definition:Dense-in-itself",
"Definition:Dense-in-itself",
"Definition:Infinite Set"
] |
proofwiki-4089 | Equal Chords in Circle | In a circle, equal chords are equally distant from the center, and chords that are equally distant from the center are equal in length.
{{:Euclid:Proposition/III/14}} | :300px
Let $ABDC$ be a circle, and let $AB$ and $CD$ be equal chords on it.
From Finding Center of Circle, let $E$ be the center of $ABDC$.
Construct $EF$, $EG$ perpendicular to $AB$ and $CD$, and join $AE$ and $EC$.
From Conditions for Diameter to be Perpendicular Bisector, $EF$ bisects $AD$.
So $AF = FB$ and so $AB =... | In a [[Definition:Circle|circle]], equal [[Definition:Chord of Circle|chords]] are equally distant from the [[Definition:Center of Circle|center]], and [[Definition:Chord of Circle|chords]] that are equally distant from the [[Definition:Center of Circle|center]] are equal in [[Definition:Length (Linear Measure)|length]... | :[[File:Euclid-III-14.png|300px]]
Let $ABDC$ be a [[Definition:Circle|circle]], and let $AB$ and $CD$ be equal [[Definition:Chord of Circle|chords]] on it.
From [[Finding Center of Circle]], let $E$ be the [[Definition:Center of Circle|center]] of $ABDC$.
Construct $EF$, $EG$ [[Definition:Perpendicular|perpendicular... | Equal Chords in Circle | https://proofwiki.org/wiki/Equal_Chords_in_Circle | https://proofwiki.org/wiki/Equal_Chords_in_Circle | [
"Circles"
] | [
"Definition:Circle",
"Definition:Circle/Chord",
"Definition:Circle/Center",
"Definition:Circle/Chord",
"Definition:Circle/Center",
"Definition:Linear Measure/Length"
] | [
"File:Euclid-III-14.png",
"Definition:Circle",
"Definition:Circle/Chord",
"Finding Center of Circle",
"Definition:Circle/Center",
"Definition:Right Angle/Perpendicular",
"Conditions for Diameter to be Perpendicular Bisector",
"Definition:Bisection",
"Definition:Quadrilateral/Square",
"Definition:Q... |
proofwiki-4090 | Relative Lengths of Chords of Circles | Of chords in a circle, the diameter is the greatest, and of the rest the nearer to the center is always greater than the more remote.
{{:Euclid:Proposition/III/15}} | :320px
Let $ABCD$ be a clrcle, let $AD$ be its diameter and $E$ the center.
Let $BC$ and $FG$ be chords of $ABCD$, where $BC$ is nearer to the center than $FG$.
Let $EH$ and $EK$ be drawn perpendicular to $BC$ and $FG$ respectively.
Because $BC$ is nearer to the center than $FG$, it follows from Book III: Definition 5 ... | Of [[Definition:Chord of Circle|chords in a circle]], the [[Definition:Diameter of Circle|diameter]] is the greatest, and of the rest the nearer to the [[Definition:Center of Circle|center]] is always greater than the more remote.
{{:Euclid:Proposition/III/15}} | :[[File:Euclid-III-15.png|320px]]
Let $ABCD$ be a [[Definition:Circle|clrcle]], let $AD$ be its [[Definition:Diameter of Circle|diameter]] and $E$ the [[Definition:Center of Circle|center]].
Let $BC$ and $FG$ be [[Definition:Chord of Circle|chords]] of $ABCD$, where $BC$ is nearer to the [[Definition:Center of Circle... | Relative Lengths of Chords of Circles | https://proofwiki.org/wiki/Relative_Lengths_of_Chords_of_Circles | https://proofwiki.org/wiki/Relative_Lengths_of_Chords_of_Circles | [
"Circles"
] | [
"Definition:Circle/Chord",
"Definition:Circle/Diameter",
"Definition:Circle/Center"
] | [
"File:Euclid-III-15.png",
"Definition:Circle",
"Definition:Circle/Diameter",
"Definition:Circle/Center",
"Definition:Circle/Chord",
"Definition:Circle/Center",
"Definition:Right Angle/Perpendicular",
"Definition:Circle/Center",
"Definition:Euclid's Definitions - Book III/5 - Greater Distance in Circ... |
proofwiki-4091 | Construction of Tangent from Point to Circle | From a given point outside a given circle, it is possible to draw a tangent to that circle.
{{:Euclid:Proposition/III/17}} | :400px
Let $A$ be the given point and let $BCD$ be the given circle.
It is required that a straight line be drawn from $A$ to $BCD$.
Let the center $E$ of $BCD$ be found.
Join $AE$ and draw the circle $AFG$ with center $E$ and radius $AE$.
From $D$ let $DF$ be drawn perpendicular to $EA$.
Join $EF$ and let $B$ be the p... | From a given [[Definition:Point|point]] outside a given [[Definition:Circle|circle]], it is possible to draw a [[Definition:Tangent to Circle|tangent]] to that [[Definition:Circle|circle]].
{{:Euclid:Proposition/III/17}} | :[[File:Euclid-III-17.png|400px]]
Let $A$ be the given [[Definition:Point|point]] and let $BCD$ be the given [[Definition:Circle|circle]].
It is required that a [[Definition:Straight Line|straight line]] be drawn from $A$ to $BCD$.
Let [[Finding Center of Circle|the center $E$ of $BCD$ be found]].
Join $AE$ and dra... | Construction of Tangent from Point to Circle/Proof 1 | https://proofwiki.org/wiki/Construction_of_Tangent_from_Point_to_Circle | https://proofwiki.org/wiki/Construction_of_Tangent_from_Point_to_Circle/Proof_1 | [
"Circles",
"Construction of Tangent from Point to Circle"
] | [
"Definition:Point",
"Definition:Circle",
"Definition:Tangent Line/Circle",
"Definition:Circle"
] | [
"File:Euclid-III-17.png",
"Definition:Point",
"Definition:Circle",
"Definition:Line/Straight Line",
"Finding Center of Circle",
"Definition:Circle",
"Definition:Circle/Radius",
"Construction of Perpendicular Line",
"Definition:Circle",
"Definition:Tangent Line/Circle",
"Definition:Circle/Center"... |
proofwiki-4092 | Construction of Tangent from Point to Circle | From a given point outside a given circle, it is possible to draw a tangent to that circle.
{{:Euclid:Proposition/III/17}} | :400px
Let $BCD$ with center $A$ be the circle, and let $E$ be the exterior point from which a tangent is to be drawn.
Bisect $AE$ at $F$.
Then draw a circle $AEG$ whose center is $F$ and whose radius is $AF$.
The point $G$ is where $AEG$ intersects $BCD$.
The line $EG$ is the required tangent.
=== Proof of Constructio... | From a given [[Definition:Point|point]] outside a given [[Definition:Circle|circle]], it is possible to draw a [[Definition:Tangent to Circle|tangent]] to that [[Definition:Circle|circle]].
{{:Euclid:Proposition/III/17}} | :[[File:Euclid-III-17a.png|400px]]
Let $BCD$ with [[Definition:Center of Circle|center]] $A$ be the [[Definition:Circle|circle]], and let $E$ be the exterior point from which a [[Definition:Tangent to Circle|tangent]] is to be drawn.
[[Bisection of Straight Line|Bisect]] $AE$ at $F$.
Then draw a [[Definition:Circle|... | Construction of Tangent from Point to Circle/Proof 2 | https://proofwiki.org/wiki/Construction_of_Tangent_from_Point_to_Circle | https://proofwiki.org/wiki/Construction_of_Tangent_from_Point_to_Circle/Proof_2 | [
"Circles",
"Construction of Tangent from Point to Circle"
] | [
"Definition:Point",
"Definition:Circle",
"Definition:Tangent Line/Circle",
"Definition:Circle"
] | [
"File:Euclid-III-17a.png",
"Definition:Circle/Center",
"Definition:Circle",
"Definition:Tangent Line/Circle",
"Bisection of Straight Line",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Definition:Line/Straight Line",
"Definition:Tangent Line/Circle",
"Definition:... |
proofwiki-4093 | Equal Angles in Equal Circles | In equal circles, equal angles stand on equal arcs, whether at the center or at the circumference of those circles.
{{:Euclid:Proposition/III/26}} | Let $ABC$ and $DEF$ be equal circles.
Let $\angle BGC = \angle EHF$ and $\angle BAC = \angle EDF$.
:500px
Let $BC$ and $EF$ be joined.
Since the circles $ABC$ and $DEF$ are equal, their radii are equal.
So $BG = EH$ and $CG = FH$.
We also have {{hypothesis}} that $\angle BGC = \angle EHF$.
So from Triangle Side-Angle-S... | In equal [[Definition:Circle|circles]], equal [[Definition:Angle|angles]] stand on equal [[Definition:Arc of Circle|arcs]], whether at the [[Definition:Center of Circle|center]] or at the [[Definition:Circumference of Circle|circumference]] of those circles.
{{:Euclid:Proposition/III/26}} | Let $ABC$ and $DEF$ be equal [[Definition:Circle|circles]].
Let $\angle BGC = \angle EHF$ and $\angle BAC = \angle EDF$.
:[[File:Euclid-III-26.png|500px]]
Let $BC$ and $EF$ be joined.
Since the [[Definition:Circle|circles]] $ABC$ and $DEF$ are equal, their [[Definition:Radius of Circle|radii]] are equal.
So $BG = ... | Equal Angles in Equal Circles | https://proofwiki.org/wiki/Equal_Angles_in_Equal_Circles | https://proofwiki.org/wiki/Equal_Angles_in_Equal_Circles | [
"Circles"
] | [
"Definition:Circle",
"Definition:Angle",
"Definition:Circle/Arc",
"Definition:Circle/Center",
"Definition:Circle/Circumference"
] | [
"Definition:Circle",
"File:Euclid-III-26.png",
"Definition:Circle",
"Definition:Circle/Radius",
"Triangle Side-Angle-Side Congruence",
"Definition:Segment of Circle",
"Definition:Segment of Circle/Similar",
"Definition:Segment of Circle",
"Definition:Segment of Circle",
"Definition:Segment of Circ... |
proofwiki-4094 | Angles on Equal Arcs are Equal | In equal circles, angles standing on equal arcs are equal to one another, whether at the center or at the circumference of those circles.
{{:Euclid:Proposition/III/27}} | Let $ABC$ and $DEF$ be equal circles.
Let $\angle BGC$ and $\angle EHF$ stand on the equal arcs $BC$ and $EF$.
:500px
Suppose $\angle BGC \ne \angle EHF$.
Then one of them is bigger.
Suppose $\angle BGC > \angle EHF$.
On the straight line $BG$, construct $\angle BGK$ equal to $\angle EHF$.
From Equal Angles in Equal Ci... | In equal [[Definition:Circle|circles]], [[Definition:Angle|angles]] standing on equal [[Definition:Arc of Circle|arcs]] are equal to one another, whether at the [[Definition:Center of Circle|center]] or at the [[Definition:Circumference of Circle|circumference]] of those [[Definition:Circle|circles]].
{{:Euclid:Propo... | Let $ABC$ and $DEF$ be equal [[Definition:Circle|circles]].
Let $\angle BGC$ and $\angle EHF$ stand on the equal [[Definition:Arc of Circle|arcs]] $BC$ and $EF$.
:[[File:Euclid-III-27.png|500px]]
Suppose $\angle BGC \ne \angle EHF$.
Then one of them is bigger.
Suppose $\angle BGC > \angle EHF$.
On the [[Definitio... | Angles on Equal Arcs are Equal | https://proofwiki.org/wiki/Angles_on_Equal_Arcs_are_Equal | https://proofwiki.org/wiki/Angles_on_Equal_Arcs_are_Equal | [
"Circles"
] | [
"Definition:Circle",
"Definition:Angle",
"Definition:Circle/Arc",
"Definition:Circle/Center",
"Definition:Circle/Circumference",
"Definition:Circle"
] | [
"Definition:Circle",
"Definition:Circle/Arc",
"File:Euclid-III-27.png",
"Definition:Line/Straight Line",
"Construction of Equal Angle",
"Equal Angles in Equal Circles",
"Definition:Angle",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"Inscribed Angle Theorem"
] |
proofwiki-4095 | Bisection of Arc | It is possible to bisect an arc of a circle.
{{:Euclid:Proposition/III/30}} | Join $AD$ and $BD$.
We have that $AC = CB$ and $CD$ is common.
We also have that $\angle ACD = \angle BCD$ as they are both right angles.
So from Triangle Side-Angle-Side Congruence $\triangle ACD = \triangle BCD$ and so $AD = BD$.
But from Straight Lines Cut Off Equal Arcs in Equal Circles, the arc $AD$ equals the arc... | It is possible to [[Definition:Bisection|bisect]] an [[Definition:Arc of Circle|arc of a circle]].
{{:Euclid:Proposition/III/30}} | Join $AD$ and $BD$.
We have that $AC = CB$ and $CD$ is common.
We also have that $\angle ACD = \angle BCD$ as they are both [[Definition:Right Angle|right angles]].
So from [[Triangle Side-Angle-Side Congruence]] $\triangle ACD = \triangle BCD$ and so $AD = BD$.
But from [[Straight Lines Cut Off Equal Arcs in Equal... | Bisection of Arc | https://proofwiki.org/wiki/Bisection_of_Arc | https://proofwiki.org/wiki/Bisection_of_Arc | [
"Circles"
] | [
"Definition:Bisection",
"Definition:Circle/Arc"
] | [
"Definition:Right Angle",
"Triangle Side-Angle-Side Congruence",
"Straight Lines Cut Off Equal Arcs in Equal Circles",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"Definition:Circle/Arc",
"Definition:Bisection"
] |
proofwiki-4096 | Relative Sizes of Angles in Segments | In a circle:
:the angle in a semicircle is right
:the angle in a segment greater than a semicircle is acute
:the angle in a segment less than a semicircle is obtuse.
Further:
:the angle of a segment greater than a semicircle is obtuse
:the angle of a segment less than a semicircle is acute.
{{:Euclid:Proposition/III/31... | :300px
Let $ABCD$ be a circle whose diameter is $BC$ and whose center is $E$.
Join $AB$, $AC$, $AD$, $DC$ and $AE$.
Let $BA$ be produced to $F$.
Since $BE = EA$, from Isosceles Triangle has Two Equal Angles it follows that $\angle ABE = \angle BAE$.
Since $CE = EA$, from Isosceles Triangle has Two Equal Angles it follo... | In a [[Definition:Circle|circle]]:
:the [[Definition:Angle in Segment|angle]] in a [[Definition:Semicircle|semicircle]] is [[Definition:Right Angle|right]]
:the [[Definition:Angle in Segment|angle]] in a [[Definition:Segment of Circle|segment]] greater than a [[Definition:Semicircle|semicircle]] is [[Definition:Acute A... | :[[File:Euclid-III-31.png|300px]]
Let $ABCD$ be a [[Definition:Circle|circle]] whose [[Definition:Diameter of Circle|diameter]] is $BC$ and whose [[Definition:Center of Circle|center]] is $E$.
Join $AB$, $AC$, $AD$, $DC$ and $AE$.
Let $BA$ be [[Definition:Production|produced]] to $F$.
Since $BE = EA$, from [[Isosce... | Relative Sizes of Angles in Segments | https://proofwiki.org/wiki/Relative_Sizes_of_Angles_in_Segments | https://proofwiki.org/wiki/Relative_Sizes_of_Angles_in_Segments | [
"Circles"
] | [
"Definition:Circle",
"Definition:Segment of Circle/Angle in Segment",
"Definition:Circle/Semicircle",
"Definition:Right Angle",
"Definition:Segment of Circle/Angle in Segment",
"Definition:Segment of Circle",
"Definition:Circle/Semicircle",
"Definition:Acute Angle",
"Definition:Segment of Circle/Ang... | [
"File:Euclid-III-31.png",
"Definition:Circle",
"Definition:Circle/Diameter",
"Definition:Circle/Center",
"Definition:Production",
"Isosceles Triangle has Two Equal Angles",
"Isosceles Triangle has Two Equal Angles",
"Sum of Angles of Triangle equals Two Right Angles",
"Definition:Euclid's Definition... |
proofwiki-4097 | Tangent-Chord Theorem | Let $EF$ be a tangent to a circle $ABCD$, touching it at $B$.
Let $BD$ be a chord of $ABCD$.
Then:
:the angle in segment $BCD$ equals $\angle DBE$
and:
:the angle in segment $BAD$ equals $\angle DBF$. | :280px
Draw $BA$ perpendicular to $EF$ through $B$.
Let $C$ be selected on the circle on the arc $BD$.
Join $AD, DC, CB$.
From Right Angle to Tangent of Circle goes through Center, the center of the circle lies on $AB$.
By definition, then, $AB$ is a diameter of the circle.
From Relative Sizes of Angles in Segments, it... | Let $EF$ be a [[Definition:Tangent to Circle|tangent]] to a [[Definition:Circle|circle]] $ABCD$, touching it at $B$.
Let $BD$ be a [[Definition:Chord of Circle|chord]] of $ABCD$.
Then:
:the [[Definition:Angle in Segment|angle]] in [[Definition:Segment of Circle|segment]] $BCD$ equals $\angle DBE$
and:
:the [[Definit... | :[[File:Euclid-III-32.png|280px]]
Draw $BA$ [[Definition:Perpendicular|perpendicular]] to $EF$ through $B$.
Let $C$ be selected on the [[Definition:Circle|circle]] on the [[Definition:Arc of Circle|arc]] $BD$.
Join $AD, DC, CB$.
From [[Right Angle to Tangent of Circle goes through Center]], the [[Definition:Center ... | Tangent-Chord Theorem | https://proofwiki.org/wiki/Tangent-Chord_Theorem | https://proofwiki.org/wiki/Tangent-Chord_Theorem | [
"Tangent-Chord Theorem",
"Circles",
"Tangents to Circles",
"Named Theorems"
] | [
"Definition:Tangent Line/Circle",
"Definition:Circle",
"Definition:Circle/Chord",
"Definition:Segment of Circle/Angle in Segment",
"Definition:Segment of Circle",
"Definition:Segment of Circle/Angle in Segment",
"Definition:Segment of Circle"
] | [
"File:Euclid-III-32.png",
"Definition:Right Angle/Perpendicular",
"Definition:Circle",
"Definition:Circle/Arc",
"Right Angle to Tangent of Circle goes through Center",
"Definition:Circle/Center",
"Definition:Circle/Diameter",
"Relative Sizes of Angles in Segments",
"Definition:Right Angle",
"Sum o... |
proofwiki-4098 | Construction of Segment on Given Line Admitting Given Angle | On any given line segment, it is possible to describe a segment of a circle which admits an angle equal to any given rectilineal angle.
{{:Euclid:Proposition/III/33}} | === Proof for Acute Angle ===
We have that $AF = FB$, $\angle AFG = \angle BFG$ (both are right angles) and $AG$ is common.
So from Triangle Side-Angle-Side Congruence it follows that $AG = BG$.
So the circle $ABE$ whose center is at $G$ and whose radius is $GB$ also passes through $A$.
As $AE$ passes through the cent... | On any given [[Definition:Line Segment|line segment]], it is possible to describe a [[Definition:Segment of Circle|segment of a circle]] which [[Definition:Angle in Segment|admits an angle]] equal to any given [[Definition:Rectilineal Angle|rectilineal angle]].
{{:Euclid:Proposition/III/33}} | === Proof for Acute Angle ===
We have that $AF = FB$, $\angle AFG = \angle BFG$ (both are [[Definition:Right Angle|right angles]]) and $AG$ is common.
So from [[Triangle Side-Angle-Side Congruence]] it follows that $AG = BG$.
So the [[Definition:Circle|circle]] $ABE$ whose [[Definition:Center of Circle|center]] is a... | Construction of Segment on Given Line Admitting Given Angle | https://proofwiki.org/wiki/Construction_of_Segment_on_Given_Line_Admitting_Given_Angle | https://proofwiki.org/wiki/Construction_of_Segment_on_Given_Line_Admitting_Given_Angle | [
"Circles"
] | [
"Definition:Line/Segment",
"Definition:Segment of Circle",
"Definition:Segment of Circle/Angle in Segment",
"Definition:Angle/Rectilineal"
] | [
"Definition:Right Angle",
"Triangle Side-Angle-Side Congruence",
"Definition:Circle",
"Definition:Circle/Center",
"Definition:Circle/Radius",
"Definition:Circle/Center",
"Definition:Circle/Diameter",
"Definition:Circle",
"Definition:Right Angle/Perpendicular",
"Line at Right Angles to Diameter of ... |
proofwiki-4099 | Tangent Secant Theorem | Let $D$ be a point outside a circle $ABC$.
Let $DB$ be tangent to the circle $ABC$.
Let $DA$ be a straight line which cuts the circle $ABC$ at $A$ and $C$.
Then $DB^2 = AD \cdot DC$.
{{:Euclid:Proposition/III/36}} | Let $DA$ pass through the center $F$ of circle $ABC$.
Join $FB$.
From Radius at Right Angle to Tangent, $\angle FBD$ is a right angle.
:300px
We have that $F$ bisects $AC$ and that $CD$ is added to it.
So we can apply Square of Sum less Square and see that:
:$AD \cdot DC + FC^2 = FD^2$
But $FC = FB$ and so:
:$AD \cdot ... | Let $D$ be a [[Definition:Point|point]] outside a [[Definition:Circle|circle]] $ABC$.
Let $DB$ be [[Definition:Tangent to Circle|tangent]] to the [[Definition:Circle|circle]] $ABC$.
Let $DA$ be a [[Definition:Straight Line|straight line]] which cuts the [[Definition:Circle|circle]] $ABC$ at $A$ and $C$.
Then $DB^2 =... | Let $DA$ pass through the [[Definition:Center of Circle|center]] $F$ of [[Definition:Circle|circle]] $ABC$.
Join $FB$.
From [[Radius at Right Angle to Tangent]], $\angle FBD$ is a [[Definition:Right Angle|right angle]].
:[[File:Euclid-III-36a.png|300px]]
We have that $F$ [[Definition:Bisect|bisects]] $AC$ and that ... | Tangent Secant Theorem/Proof 1 | https://proofwiki.org/wiki/Tangent_Secant_Theorem | https://proofwiki.org/wiki/Tangent_Secant_Theorem/Proof_1 | [
"Tangent Secant Theorem",
"Circles",
"Tangents to Circles",
"Named Theorems"
] | [
"Definition:Point",
"Definition:Circle",
"Definition:Tangent Line/Circle",
"Definition:Circle",
"Definition:Line/Straight Line",
"Definition:Circle"
] | [
"Definition:Circle/Center",
"Definition:Circle",
"Radius at Right Angle to Tangent",
"Definition:Right Angle",
"File:Euclid-III-36a.png",
"Definition:Bisection",
"Square of Sum less Square",
"Pythagoras's Theorem",
"Definition:Circle/Center",
"Definition:Circle",
"Definition:Right Angle/Perpendi... |
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