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-3700 | Compact Space in Particular Point Space | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $\set p$ is compact in $T$. | Any open cover of $\set p$ has a finite subcover: an arbitrary single set that contains $p$ is a cover for $\set p$.
So $\set p$ is compact in $T$.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $\set p$ is [[Definition:Compact Topological Subspace|compact]] in $T$. | Any [[Definition:Open Cover|open cover]] of $\set p$ has a [[Definition:Finite Subcover|finite subcover]]: an arbitrary single [[Definition:Set|set]] that contains $p$ is a [[Definition:Cover of Set|cover]] for $\set p$.
So $\set p$ is [[Definition:Compact Topological Subspace|compact]] in $T$.
{{qed}} | Compact Space in Particular Point Space | https://proofwiki.org/wiki/Compact_Space_in_Particular_Point_Space | https://proofwiki.org/wiki/Compact_Space_in_Particular_Point_Space | [
"Particular Point Topologies",
"Examples of Compact Topological Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Compact Topological Space/Subspace"
] | [
"Definition:Open Cover",
"Definition:Subcover/Finite",
"Definition:Set",
"Definition:Cover of Set",
"Definition:Compact Topological Space/Subspace"
] |
proofwiki-3701 | Finite Multiplicative Subgroup of Field is Cyclic | Let $\struct {F, +, \times}$ be a field.
Let $\struct {F^*, \times}$ denote the multiplicative group of $F$.
Let $C$ be a finite subgroup of $\struct {F^*, \times}$.
Then $C$ is cyclic. | We have {{afortiori}} that $\struct {F^*, \times}$ is an abelian group.
From Subgroup of Abelian Group is Abelian, $C$ is a finite abelian group.
From Fundamental Theorem of Finite Abelian Groups, $C$ is the internal group direct product of cyclic groups $H_1, H_2, \ldots, H_r$ whose orders are given as:
:$\order {H_i}... | Let $\struct {F, +, \times}$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $\struct {F^*, \times}$ denote the [[Definition:Multiplicative Group|multiplicative group]] of $F$.
Let $C$ be a [[Definition:Finite Group|finite]] [[Definition:Subgroup|subgroup]] of $\struct {F^*, \times}$.
Then $C$ is [[Definiti... | We have {{afortiori}} that $\struct {F^*, \times}$ is an [[Definition:Abelian Group|abelian group]].
From [[Subgroup of Abelian Group is Abelian]], $C$ is a [[Definition:Finite Group|finite]] [[Definition:Abelian Group|abelian group]].
From [[Fundamental Theorem of Finite Abelian Groups]], $C$ is the [[Definition:Int... | Finite Multiplicative Subgroup of Field is Cyclic | https://proofwiki.org/wiki/Finite_Multiplicative_Subgroup_of_Field_is_Cyclic | https://proofwiki.org/wiki/Finite_Multiplicative_Subgroup_of_Field_is_Cyclic | [
"Finite Multiplicative Subgroup of Field is Cyclic",
"Field Theory",
"Cyclic Groups"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Multiplicative Group",
"Definition:Finite Group",
"Definition:Subgroup",
"Definition:Cyclic Group"
] | [
"Definition:Abelian Group",
"Subgroup of Abelian Group is Abelian",
"Definition:Finite Group",
"Definition:Abelian Group",
"Fundamental Theorem of Finite Abelian Groups",
"Definition:Internal Group Direct Product",
"Definition:Cyclic Group",
"Definition:Order of Structure",
"Definition:Prime Number"... |
proofwiki-3702 | Induced Homomorphism of Polynomial Forms | Let $R$ and $S$ be commutative rings with unity.
Let $\phi: R \to S$ be a ring homomorphism.
Let $R \sqbrk X$ and $S \sqbrk X$ be the rings of polynomial forms over $R$ and $S$ respectively in the indeterminate $X$.
Then the map $\overline \phi: R \sqbrk X \to S \sqbrk X$ given by:
:$\map {\overline \phi} {a_0 + a_1 X ... | Let:
:$f = a_0 + \cdots + a_n X^n$
:$g = b_0 + \cdots + b_m X^m \in R \sqbrk X$
We have:
{{begin-eqn}}
{{eqn | l = \map {\overline \phi} f \map {\overline \phi} g
| r = \sqbrk {\map \phi {a_0} + \cdots + \map \phi {a_n} X^n} \sqbrk {\map \phi {b_0} + \cdots + \map \phi {b_m} X^m}
| c =
}}
{{eqn | r = \sum_... | Let $R$ and $S$ be [[Definition:Commutative and Unitary Ring|commutative rings with unity]].
Let $\phi: R \to S$ be a [[Definition:Ring Homomorphism|ring homomorphism]].
Let $R \sqbrk X$ and $S \sqbrk X$ be the [[Definition:Ring of Polynomial Forms|rings of polynomial forms]] over $R$ and $S$ respectively in the [[De... | Let:
:$f = a_0 + \cdots + a_n X^n$
:$g = b_0 + \cdots + b_m X^m \in R \sqbrk X$
We have:
{{begin-eqn}}
{{eqn | l = \map {\overline \phi} f \map {\overline \phi} g
| r = \sqbrk {\map \phi {a_0} + \cdots + \map \phi {a_n} X^n} \sqbrk {\map \phi {b_0} + \cdots + \map \phi {b_m} X^m}
| c =
}}
{{eqn | r = \sum... | Induced Homomorphism of Polynomial Forms/Proof 2 | https://proofwiki.org/wiki/Induced_Homomorphism_of_Polynomial_Forms | https://proofwiki.org/wiki/Induced_Homomorphism_of_Polynomial_Forms/Proof_2 | [
"Induced Homomorphism of Polynomial Forms",
"Polynomial Theory"
] | [
"Definition:Commutative and Unitary Ring",
"Definition:Ring Homomorphism",
"Definition:Ring of Polynomial Forms",
"Definition:Indeterminate",
"Definition:Ring Homomorphism"
] | [
"Definition:Ring Homomorphism",
"Definition:Ring Homomorphism",
"Definition:Ring Homomorphism"
] |
proofwiki-3703 | Structure of Simple Algebraic Field Extension | Let $F / K$ be a field extension.
Let $\alpha \in F$ be algebraic over $K$.
Let $\mu_\alpha$ be the minimal polynomial of $\alpha$ over $K$.
Let $K \sqbrk X$ be the ring of polynomial functions.
Let $\gen {\mu_\alpha}$ be the ideal of $K \sqbrk X$ generated by $\mu_\alpha$.
Let $K \sqbrk \alpha$ be the subring of $F$ g... | Define $\phi: K \sqbrk X \to K \sqbrk \alpha$ by:
:$\map \phi f = \map f \alpha$
We have
:$\map \phi f = 0 \iff \map f \alpha = 0 \iff \mu_\alpha \divides f$
where the last equivalence follows from $\mu_\alpha$ being the minimal polynomial of $\alpha$.
Thus:
:$\ker \phi = \set {f \in K \sqbrk X: \mu_\alpha \divides f} ... | Let $F / K$ be a [[Definition:Field Extension|field extension]].
Let $\alpha \in F$ be [[Definition:Algebraic Field Extension|algebraic]] over $K$.
Let $\mu_\alpha$ be the [[Definition:Minimal Polynomial|minimal polynomial]] of $\alpha$ over $K$.
Let $K \sqbrk X$ be the [[Definition:Ring of Polynomial Functions|ring... | Define $\phi: K \sqbrk X \to K \sqbrk \alpha$ by:
:$\map \phi f = \map f \alpha$
We have
:$\map \phi f = 0 \iff \map f \alpha = 0 \iff \mu_\alpha \divides f$
where the last equivalence follows from $\mu_\alpha$ being the [[Definition:Minimal Polynomial/Definition 3|minimal polynomial]] of $\alpha$.
Thus:
:$\ker \phi ... | Structure of Simple Algebraic Field Extension | https://proofwiki.org/wiki/Structure_of_Simple_Algebraic_Field_Extension | https://proofwiki.org/wiki/Structure_of_Simple_Algebraic_Field_Extension | [
"Field Extensions"
] | [
"Definition:Field Extension",
"Definition:Algebraic Field Extension",
"Definition:Minimal Polynomial",
"Definition:Ring of Polynomial Functions",
"Definition:Ideal of Ring",
"Definition:Generator of Ideal of Ring",
"Definition:Subring",
"Definition:Generator of Field",
"Definition:Subfield",
"Defi... | [
"Definition:Minimal Polynomial/Definition 3",
"Definition:Surjection",
"First Isomorphism Theorem/Rings",
"Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal",
"Definition:Maximal Ideal of Ring",
"Maximal Ideal iff Quotient Ring is Field",
"Field is Integral Domain",
"Fiel... |
proofwiki-3704 | Subset of Particular Point Space is either Open or Closed | Let $T = \struct {S, \tau_p}$ be a particular point space.
Let $H \subseteq S$ be any subset of $T$.
Then $H$ is either open or closed in $T$.
The only sets which are both closed and open in $T$ are $S$ and $\O$. | Let $H \subseteq S$.
There are two cases to consider:
:$p \in H$
:$p \notin H$
If $p \in H$ then by definition of a particular point topology, $H$ is open.
If $p \notin H$, then $p \in \relcomp S H$, where $\relcomp S H$ is the relative complement of $H$ in $S$.
So $\relcomp S H$ is open by definition of a particular p... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Let $H \subseteq S$ be any [[Definition:Subset|subset]] of $T$.
Then $H$ is either [[Definition:Open Set (Topology)|open]] or [[Definition:Closed Set (Topology)|closed]] in $T$.
The only sets which are [[Definition:... | Let $H \subseteq S$.
There are two cases to consider:
:$p \in H$
:$p \notin H$
If $p \in H$ then by definition of a [[Definition:Particular Point Topology|particular point topology]], $H$ is [[Definition:Open Set (Topology)|open]].
If $p \notin H$, then $p \in \relcomp S H$, where $\relcomp S H$ is the [[Definitio... | Subset of Particular Point Space is either Open or Closed | https://proofwiki.org/wiki/Subset_of_Particular_Point_Space_is_either_Open_or_Closed | https://proofwiki.org/wiki/Subset_of_Particular_Point_Space_is_either_Open_or_Closed | [
"Particular Point Topologies",
"Open Sets",
"Closed Sets",
"Clopen Sets"
] | [
"Definition:Particular Point Topology",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Clopen Set"
] | [
"Definition:Particular Point Topology",
"Definition:Open Set/Topology",
"Definition:Relative Complement",
"Definition:Open Set/Topology",
"Definition:Particular Point Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Relative Complement of Relative Complement",
"Definit... |
proofwiki-3705 | Particular Point Space is Weakly Locally Compact | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is weakly locally compact. | By Compact Space in Particular Point Space, $\set p$ is compact in $T$.
But by definition $\set p$ is open in $T$.
So $p$ is contained in a compact neighborhood, that is, $\set p$.
Now let $x \in S: x \ne p$.
Then $\set {x, p}$ is open in $T$.
We have that $\set {x, p}$ trivially has an open cover, that is, $\set {\set... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $T$ is [[Definition:Weakly Locally Compact Space|weakly locally compact]]. | By [[Compact Space in Particular Point Space]], $\set p$ is compact in $T$.
But by definition $\set p$ is [[Definition:Open Set (Topology)|open]] in $T$.
So $p$ is contained in a [[Definition:Compact Topological Subspace|compact]] [[Definition:Neighborhood of Point|neighborhood]], that is, $\set p$.
Now let $x \in ... | Particular Point Space is Weakly Locally Compact | https://proofwiki.org/wiki/Particular_Point_Space_is_Weakly_Locally_Compact | https://proofwiki.org/wiki/Particular_Point_Space_is_Weakly_Locally_Compact | [
"Particular Point Topologies",
"Examples of Weakly Locally Compact Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Weakly Locally Compact Space"
] | [
"Compact Space in Particular Point Space",
"Definition:Open Set/Topology",
"Definition:Compact Topological Space/Subspace",
"Definition:Neighborhood (Topology)/Point",
"Definition:Open Set/Topology",
"Definition:Open Cover",
"Definition:Subcover/Finite",
"Definition:Set",
"Definition:Neighborhood (T... |
proofwiki-3706 | Infinite Particular Point Space is not Strongly Locally Compact | Let $T = \struct {S, \tau_p}$ be an infinite particular point space.
Then $T$ is not strongly locally compact. | By definition, $T$ is strongly locally compact {{iff}} every point of $S$ is contained in an open set whose closure is compact.
Let $x \in S: x \ne p$.
Let $x \in U$ where $U$ is open in $T$.
From Closure of Open Set of Particular Point Space we have:
:$U^- = S$
where $U^-$ is the closure of $U$.
But from Infinite Part... | Let $T = \struct {S, \tau_p}$ be an [[Definition:Infinite Particular Point Topology|infinite particular point space]].
Then $T$ is not [[Definition:Strongly Locally Compact Space|strongly locally compact]]. | By definition, $T$ is [[Definition:Strongly Locally Compact Space|strongly locally compact]] {{iff}} every [[Definition:Point of Set|point]] of $S$ is contained in an [[Definition:Open Set (Topology)|open set]] whose [[Definition:Closure (Topology)|closure]] is [[Definition:Compact Topological Subspace|compact]].
Let... | Infinite Particular Point Space is not Strongly Locally Compact | https://proofwiki.org/wiki/Infinite_Particular_Point_Space_is_not_Strongly_Locally_Compact | https://proofwiki.org/wiki/Infinite_Particular_Point_Space_is_not_Strongly_Locally_Compact | [
"Infinite Particular Point Topologies",
"Examples of Strongly Locally Compact Spaces"
] | [
"Definition:Particular Point Topology/Infinite",
"Definition:Strongly Locally Compact Space"
] | [
"Definition:Strongly Locally Compact Space",
"Definition:Element",
"Definition:Open Set/Topology",
"Definition:Closure (Topology)",
"Definition:Compact Topological Space/Subspace",
"Definition:Open Set/Topology",
"Closure of Open Set of Particular Point Space",
"Definition:Closure (Topology)",
"Infi... |
proofwiki-3707 | Infinite Particular Point Space is not Compact | Let $T = \struct {S, \tau_p}$ be an infinite particular point space.
Then $T$ is not compact. | Consider the open cover of $T$:
:$\CC = \set {\set {x, p}: x \in S, x \ne p}$
As $S$ is infinite, then so is $\CC$, as we can set up a bijection from $\phi: S \setminus \set p \leftrightarrow \CC$:
:$\forall x \in S \setminus \set p: \map \phi x = \set {x, p}$
Hence $\CC$ has no finite subcover.
The result follows by d... | Let $T = \struct {S, \tau_p}$ be an [[Definition:Infinite Particular Point Topology|infinite particular point space]].
Then $T$ is not [[Definition:Compact Topological Space|compact]]. | Consider the [[Definition:Open Cover|open cover]] of $T$:
:$\CC = \set {\set {x, p}: x \in S, x \ne p}$
As $S$ is [[Definition:Infinite|infinite]], then so is $\CC$, as we can set up a [[Definition:Bijection|bijection]] from $\phi: S \setminus \set p \leftrightarrow \CC$:
:$\forall x \in S \setminus \set p: \map \phi ... | Infinite Particular Point Space is not Compact | https://proofwiki.org/wiki/Infinite_Particular_Point_Space_is_not_Compact | https://proofwiki.org/wiki/Infinite_Particular_Point_Space_is_not_Compact | [
"Particular Point Topologies",
"Examples of Compact Topological Spaces"
] | [
"Definition:Particular Point Topology/Infinite",
"Definition:Compact Topological Space"
] | [
"Definition:Open Cover",
"Definition:Infinite",
"Definition:Bijection",
"Definition:Subcover/Finite",
"Definition:Compact Topological Space"
] |
proofwiki-3708 | Uncountable Particular Point Space is not Lindelöf | Let $T = \struct {S, \tau_p}$ be an uncountable particular point space.
Then $T$ is not a Lindelöf space. | Consider the open cover of $T$:
:$\CC = \set {\set {x, p}: x \in S, x \ne p}$
As $S$ is uncountable, then so is $\CC$, as we can set up a bijection $\phi: S \setminus \set p \leftrightarrow \CC$:
:$\forall x \in S \setminus \set p: \map \phi x = \set {x, p}$
Hence $\CC$ has no countable subcover.
The result follows by ... | Let $T = \struct {S, \tau_p}$ be an [[Definition:Uncountable Particular Point Topology|uncountable particular point space]].
Then $T$ is not a [[Definition:Lindelöf Space|Lindelöf space]]. | Consider the [[Definition:Open Cover|open cover]] of $T$:
:$\CC = \set {\set {x, p}: x \in S, x \ne p}$
As $S$ is [[Definition:Uncountable Set|uncountable]], then so is $\CC$, as we can set up a [[Definition:Bijection|bijection]] $\phi: S \setminus \set p \leftrightarrow \CC$:
:$\forall x \in S \setminus \set p: \map ... | Uncountable Particular Point Space is not Lindelöf | https://proofwiki.org/wiki/Uncountable_Particular_Point_Space_is_not_Lindelöf | https://proofwiki.org/wiki/Uncountable_Particular_Point_Space_is_not_Lindelöf | [
"Uncountable Particular Point Topologies",
"Examples of Lindelöf Spaces"
] | [
"Definition:Particular Point Topology/Uncountable",
"Definition:Lindelöf Space"
] | [
"Definition:Open Cover",
"Definition:Uncountable/Set",
"Definition:Bijection",
"Definition:Subcover/Countable",
"Definition:Lindelöf Space"
] |
proofwiki-3709 | Countable Particular Point Space is Lindelöf | Let $T = \struct {S, \tau_p}$ be a countable particular point space.
Then $T$ is a Lindelöf apace. | Consider the open cover of $T$:
:$\CC = \set {\set {x, p}: x \in S, x \ne p}$
As $S$ is countable, then so is $\CC$, as we can set up a bijection from $\phi: S \setminus \set p \leftrightarrow \CC$:
:$\forall x \in S \setminus \set p: \map \phi x = \set {x, p}$
Hence $\CC$ is its own countable subcover.
The result foll... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Countable Particular Point Topology|countable particular point space]].
Then $T$ is a [[Definition:Lindelöf Space|Lindelöf apace]]. | Consider the [[Definition:Open Cover|open cover]] of $T$:
:$\CC = \set {\set {x, p}: x \in S, x \ne p}$
As $S$ is [[Definition:Countable|countable]], then so is $\CC$, as we can set up a [[Definition:Bijection|bijection]] from $\phi: S \setminus \set p \leftrightarrow \CC$:
:$\forall x \in S \setminus \set p: \map \ph... | Countable Particular Point Space is Lindelöf | https://proofwiki.org/wiki/Countable_Particular_Point_Space_is_Lindelöf | https://proofwiki.org/wiki/Countable_Particular_Point_Space_is_Lindelöf | [
"Countable Particular Point Topologies",
"Examples of Lindelöf Spaces"
] | [
"Definition:Particular Point Topology/Countable",
"Definition:Lindelöf Space"
] | [
"Definition:Open Cover",
"Definition:Countable Set",
"Definition:Bijection",
"Definition:Subcover/Countable",
"Definition:Lindelöf Space"
] |
proofwiki-3710 | Particular Point Space is Separable | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is separable. | By definition, $T$ is separable {{iff}} there exists a countable subset of $S$ which is everywhere dense in $T$.
Consider $U := \set p \subseteq S$.
By definition, $U$ is open in $T$.
From Closure of Open Set of Particular Point Space we have that $U^- = S$, where $U^-$ is the closure of $U$.
By definition, $U$ is ever... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $T$ is [[Definition:Separable Space|separable]]. | By definition, $T$ is [[Definition:Separable Space|separable]] {{iff}} there exists a [[Definition:Countable Set|countable]] [[Definition:Subset|subset]] of $S$ which is [[Definition:Everywhere Dense|everywhere dense]] in $T$.
Consider $U := \set p \subseteq S$.
By definition, $U$ is [[Definition:Open Set (Topology)... | Particular Point Space is Separable | https://proofwiki.org/wiki/Particular_Point_Space_is_Separable | https://proofwiki.org/wiki/Particular_Point_Space_is_Separable | [
"Particular Point Topologies",
"Examples of Separable Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Separable Space"
] | [
"Definition:Separable Space",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Everywhere Dense",
"Definition:Open Set/Topology",
"Closure of Open Set of Particular Point Space",
"Definition:Closure (Topology)",
"Definition:Everywhere Dense",
"Definition:Countable Set",
"Definition:Sep... |
proofwiki-3711 | Field Adjoined Set | Let $F$ be a field.
Let $S \subseteq F$ be a subset of $F$.
Let $K \le F$ be a subfield of $F$.
The subring $K \sqbrk S$ of $F$ generated by $K \cup S$ is the set of all finite linear combinations of powers of elements of $S$ with coefficients in $K$.
The subfield $\map K S$ of $F$ generated by $K \cup S$ is the set of... | {{explain|Most of this needs to be explained in more detail. Suffers from coherence issues.}}
Let $\set {X_s: s \in S}$ be a family of indeterminates indexed by $S$.
Let $\phi$ be the Evaluation Homomorphism such that $\phi \sqbrk {X_s} = s$.
{{explain|Can the above Evaluation Homomorphism link be replaced by Definitio... | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $S \subseteq F$ be a [[Definition:Subset|subset]] of $F$.
Let $K \le F$ be a [[Definition:Subfield|subfield]] of $F$.
The [[Definition:Subring|subring]] $K \sqbrk S$ of $F$ [[Definition:Generator of Subfield|generated]] by $K \cup S$ is the [[Definitio... | {{explain|Most of this needs to be explained in more detail. Suffers from coherence issues.}}
Let $\set {X_s: s \in S}$ be a family of indeterminates [[Definition:Indexing Set|indexed]] by $S$.
Let $\phi$ be the [[Evaluation Homomorphism]] such that $\phi \sqbrk {X_s} = s$.
{{explain|Can the above [[Evaluation Homom... | Field Adjoined Set | https://proofwiki.org/wiki/Field_Adjoined_Set | https://proofwiki.org/wiki/Field_Adjoined_Set | [
"Field Extensions"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Subset",
"Definition:Subfield",
"Definition:Subring",
"Definition:Generator of Field",
"Definition:Set",
"Definition:Finite Set",
"Definition:Linear Combination",
"Definition:Subfield",
"Definition:Generator of Field",
"Definition:Set",
"Defin... | [
"Definition:Indexing Set",
"Universal Property of Polynomial Ring",
"Universal Property of Polynomial Ring",
"Definition:Polynomial Evaluation Homomorphism",
"Ring Homomorphism Preserves Subrings",
"Definition:Polynomial",
"Definition:Finite",
"Definition:Linear Combination",
"Definition:Closure (Ab... |
proofwiki-3712 | Separability in Uncountable Particular Point Space | Let $T = \struct {S, \tau_p}$ be an uncountable particular point space.
Let $H = S \setminus \set p$ where $\setminus$ denotes set difference.
Then $H$ is not separable. | By definition, $H$ is separable {{iff}} there exists a countable subset of $S$ which is everywhere dense in $T$.
Let $V \subseteq H$ where $V$ is countable.
$V$ is not open in $T$ as it does not contain $p$.
From Subset of Particular Point Space is either Open or Closed it follows that $V$ is closed.
From Closed Set Eq... | Let $T = \struct {S, \tau_p}$ be an [[Definition:Uncountable Particular Point Topology|uncountable particular point space]].
Let $H = S \setminus \set p$ where $\setminus$ denotes [[Definition:Set Difference|set difference]].
Then $H$ is not [[Definition:Separable Space|separable]]. | By definition, $H$ is [[Definition:Separable Space|separable]] {{iff}} there exists a [[Definition:Countable Set|countable]] [[Definition:Subset|subset]] of $S$ which is [[Definition:Everywhere Dense|everywhere dense]] in $T$.
Let $V \subseteq H$ where $V$ is [[Definition:Countable Set|countable]].
$V$ is not [[Defi... | Separability in Uncountable Particular Point Space | https://proofwiki.org/wiki/Separability_in_Uncountable_Particular_Point_Space | https://proofwiki.org/wiki/Separability_in_Uncountable_Particular_Point_Space | [
"Uncountable Particular Point Topologies",
"Examples of Separable Spaces"
] | [
"Definition:Particular Point Topology/Uncountable",
"Definition:Set Difference",
"Definition:Separable Space"
] | [
"Definition:Separable Space",
"Definition:Countable Set",
"Definition:Subset",
"Definition:Everywhere Dense",
"Definition:Countable Set",
"Definition:Open Set/Topology",
"Subset of Particular Point Space is either Open or Closed",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topologic... |
proofwiki-3713 | Uncountable Particular Point Space is not Second-Countable | Let $T = \struct {S, \tau_p}$ be an uncountable particular point space.
Then $T$ is not second-countable. | Let $H = S \setminus \set p$ where $\setminus$ denotes set difference.
Every subset $V \subseteq H$ is a closed set from Subset of Particular Point Space is either Open or Closed.
Thus we can consider $H$ as an uncountable discrete space.
The result follows from Uncountable Discrete Space is not Second-Countable.
{{qed... | Let $T = \struct {S, \tau_p}$ be an [[Definition:Uncountable Particular Point Topology|uncountable particular point space]].
Then $T$ is not [[Definition:Second-Countable Space|second-countable]]. | Let $H = S \setminus \set p$ where $\setminus$ denotes [[Definition:Set Difference|set difference]].
Every subset $V \subseteq H$ is a [[Definition:Closed Set (Topology)|closed set]] from [[Subset of Particular Point Space is either Open or Closed]].
Thus we can consider $H$ as an [[Definition:Uncountable Discrete To... | Uncountable Particular Point Space is not Second-Countable | https://proofwiki.org/wiki/Uncountable_Particular_Point_Space_is_not_Second-Countable | https://proofwiki.org/wiki/Uncountable_Particular_Point_Space_is_not_Second-Countable | [
"Uncountable Particular Point Topologies",
"Examples of Second-Countable Spaces"
] | [
"Definition:Particular Point Topology/Uncountable",
"Definition:Second-Countable Space"
] | [
"Definition:Set Difference",
"Definition:Closed Set/Topology",
"Subset of Particular Point Space is either Open or Closed",
"Definition:Discrete Topology/Uncountable",
"Uncountable Discrete Space is not Second-Countable"
] |
proofwiki-3714 | Particular Point Space is First-Countable | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is first-countable. | Let $x \in S: x \ne p$.
Consider the set $U_x = \set {x, p} \subseteq S$.
Now let $V \in \tau_p$ be an open set in $S$ such that $x \in V$.
So $x \in V$, by definition of $V$, and $p \in V$ as $V$ is open.
It follows directly that $U_x \subseteq V$.
So $\set {U_x}$ is a local basis at $x$ which is (trivially) countable... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $T$ is [[Definition:First-Countable Space|first-countable]]. | Let $x \in S: x \ne p$.
Consider the set $U_x = \set {x, p} \subseteq S$.
Now let $V \in \tau_p$ be an [[Definition:Open Set (Topology)|open set]] in $S$ such that $x \in V$.
So $x \in V$, by definition of $V$, and $p \in V$ as $V$ is [[Definition:Open Set (Topology)|open]].
It follows directly that $U_x \subseteq ... | Particular Point Space is First-Countable | https://proofwiki.org/wiki/Particular_Point_Space_is_First-Countable | https://proofwiki.org/wiki/Particular_Point_Space_is_First-Countable | [
"Particular Point Topologies",
"Examples of First-Countable Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:First-Countable Space"
] | [
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Local Basis",
"Definition:Countable Set",
"Definition:Countable Set",
"Definition:Local Basis",
"Definition:First-Countable Space"
] |
proofwiki-3715 | Kronecker’s Theorem | Let $K$ be a field.
Let $f$ be a polynomial over $K$ of degree $n \ge 1$.
Then there exists a finite extension $L / K$ of $K$ such that $f$ has at least one root in $L$.
Moreover, we can choose $L$ such that the degree $\index L K$ of $L / K$ satisfies $\index L K \le n$.
{{explain|Work out exactly which definitions of... | Let $K \sqbrk X$ be the ring of polynomial forms over $K$.
By Polynomial Forms over Field form Unique Factorization Domain, $K \sqbrk X$ is a unique factorisation domain.
Therefore, we can write $f = u g_1 \cdots g_r$, where $u$ a unit of $K \sqbrk X$ and $g_i$ is irreducible for $i = 1, \ldots, r$.
Clearly it is suffi... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $f$ be a [[Definition:Polynomial (Abstract Algebra)|polynomial]] over $K$ of [[Definition:Degree (Polynomial)|degree]] $n \ge 1$.
Then there exists a [[Definition:Finite Field Extension|finite extension]] $L / K$ of $K$ such that $f$ has at least one [[D... | Let $K \sqbrk X$ be the [[Definition:Ring of Polynomial Forms|ring of polynomial forms]] over $K$.
By [[Polynomial Forms over Field form Unique Factorization Domain]], $K \sqbrk X$ is a [[Definition:Unique Factorization Domain|unique factorisation domain]].
Therefore, we can write $f = u g_1 \cdots g_r$, where $u$ a ... | Kronecker’s Theorem | https://proofwiki.org/wiki/Kronecker’s_Theorem | https://proofwiki.org/wiki/Kronecker’s_Theorem | [
"Field Extensions"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Polynomial over Ring",
"Definition:Degree of Polynomial",
"Definition:Field Extension/Degree/Finite",
"Definition:Root of Polynomial",
"Definition:Field Extension/Degree"
] | [
"Definition:Ring of Polynomial Forms",
"Polynomial Forms over Field form Principal Ideal Domain/Corollary 3",
"Definition:Unique Factorization Domain",
"Definition:Unit of Ring",
"Definition:Irreducible Polynomial",
"Definition:Field Extension",
"Definition:Generated Ideal of Ring",
"Principal Ideal o... |
proofwiki-3716 | Homeomorphic Non-Comparable Particular Point Topologies | Let $S$ be a set with at least two elements.
Let $p, q \in S: p \ne q$.
Let $\tau_p$ and $\tau_q$ be the particular point topologies on $S$ by $p$ and $q$ respectively.
Then the topological spaces $T_p = \struct {S, \tau_p}$ and $T_q = \struct {S, \tau_q}$ are homeomorphic.
However, $\tau_p$ and $\tau_q$ are not compar... | We can set up the mapping $\phi: S \to S$:
:<nowiki>$\forall x \in S: \map \phi x = \begin {cases}
q & : x = p \\
p & : x = q \\
x & : \text {otherwise} \end {cases}$</nowiki>
It is straightforward to show that $\phi$ is a homeomorphism.
However, we have, for example, that $\set q \notin \tau_p$ and $\set p \notin \tau... | Let $S$ be a [[Definition:Set|set]] with at least two [[Definition:Element|elements]].
Let $p, q \in S: p \ne q$.
Let $\tau_p$ and $\tau_q$ be the [[Definition:Particular Point Topology|particular point topologies]] on $S$ by $p$ and $q$ respectively.
Then the [[Definition:Topological Space|topological spaces]] $T_... | We can set up the [[Definition:Mapping|mapping]] $\phi: S \to S$:
:<nowiki>$\forall x \in S: \map \phi x = \begin {cases}
q & : x = p \\
p & : x = q \\
x & : \text {otherwise} \end {cases}$</nowiki>
It is straightforward to show that $\phi$ is a [[Definition:Homeomorphism (Topological Spaces)|homeomorphism]].
Howeve... | Homeomorphic Non-Comparable Particular Point Topologies | https://proofwiki.org/wiki/Homeomorphic_Non-Comparable_Particular_Point_Topologies | https://proofwiki.org/wiki/Homeomorphic_Non-Comparable_Particular_Point_Topologies | [
"Particular Point Topologies",
"Homeomorphisms (Topological Spaces)",
"Examples of Comparable Topologies"
] | [
"Definition:Set",
"Definition:Element",
"Definition:Particular Point Topology",
"Definition:Topological Space",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Comparable Topologies"
] | [
"Definition:Mapping",
"Definition:Homeomorphism/Topological Spaces",
"Definition:Subset",
"Definition:Comparable Topologies"
] |
proofwiki-3717 | Universal Property for Simple Field Extensions | Let $F / K$ be a field extension.
Let $\alpha \in F$ be an algebraic over $K$.
Let $\mu_\alpha$ be the minimal polynomial over $\alpha$ over $K$.
Let $\psi : K \left({\alpha}\right) \to L$ be a homomorphism.
Let $\phi = \psi \restriction_K$.
Let $\overline \phi: K \left[{X}\right] \to L \left[{X}\right]$ be the Induced... | Let $\psi: K \left({\alpha}\right) \to L$ be a homomorphism.
Let $\phi = \psi \big|_K$.
Let $\overline \phi: K \left[{X}\right] \to L \left[{X}\right]$ be the Induced Homomorphism of Polynomial Forms.
For any $f = a_0 + \dotsb + a_n X^n \in K \left[{X}\right]$ we have:
{{begin-eqn}}
{{eqn | l = \psi \left({f \left({\al... | Let $F / K$ be a [[Definition:Field Extension|field extension]].
Let $\alpha \in F$ be an [[Definition:Algebraic Field Extension|algebraic]] over $K$.
Let $\mu_\alpha$ be the [[Definition:Minimal Polynomial|minimal polynomial]] over $\alpha$ over $K$.
Let $\psi : K \left({\alpha}\right) \to L$ be a [[Definition:Fie... | Let $\psi: K \left({\alpha}\right) \to L$ be a [[Definition:Field Homomorphism|homomorphism]].
Let $\phi = \psi \big|_K$.
Let $\overline \phi: K \left[{X}\right] \to L \left[{X}\right]$ be the [[Induced Homomorphism of Polynomial Forms]].
For any $f = a_0 + \dotsb + a_n X^n \in K \left[{X}\right]$ we have:
{{begin-... | Universal Property for Simple Field Extensions | https://proofwiki.org/wiki/Universal_Property_for_Simple_Field_Extensions | https://proofwiki.org/wiki/Universal_Property_for_Simple_Field_Extensions | [
"Field Extensions",
"Universal Properties"
] | [
"Definition:Field Extension",
"Definition:Algebraic Field Extension",
"Definition:Minimal Polynomial",
"Definition:Field Homomorphism",
"Induced Homomorphism of Polynomial Forms",
"Definition:Root of Polynomial",
"Definition:Field (Abstract Algebra)",
"Definition:Field Homomorphism",
"Definition:Roo... | [
"Definition:Field Homomorphism",
"Induced Homomorphism of Polynomial Forms",
"Ring Homomorphism Preserves Zero",
"Structure of Simple Algebraic Field Extension",
"Definition:Isomorphism (Abstract Algebra)/Field Isomorphism",
"Definition:Identity Mapping",
"Universal Property of Polynomial Ring",
"Univ... |
proofwiki-3718 | Finite Field Extension is Algebraic | Let $L / K$ be a finite field extension.
Then $L / K$ is algebraic. | Let $x \in L$ be arbitrary.
Let $n = \index L K$ be the degree of $L$ over $K$.
From Size of Linearly Independent Subset is at Most Size of Finite Generator, there is a $K$-linear combination of $\set {1, \ldots, x^n}$ equal to $0$.
Say $a_n x^n + \cdots + a_1 x + a_0 = 0$, $a_i \in K$, $i = 0, \ldots, n$.
Therefore $x... | Let $L / K$ be a [[Definition:Finite Field Extension|finite field extension]].
Then $L / K$ is [[Definition:Algebraic Field Extension|algebraic]]. | Let $x \in L$ be arbitrary.
Let $n = \index L K$ be the [[Definition:Degree of Field Extension|degree]] of $L$ over $K$.
From [[Size of Linearly Independent Subset is at Most Size of Finite Generator]], there is a $K$-[[Definition:Linear Combination|linear combination]] of $\set {1, \ldots, x^n}$ equal to $0$.
Say $... | Finite Field Extension is Algebraic | https://proofwiki.org/wiki/Finite_Field_Extension_is_Algebraic | https://proofwiki.org/wiki/Finite_Field_Extension_is_Algebraic | [
"Field Extensions"
] | [
"Definition:Field Extension/Degree/Finite",
"Definition:Algebraic Field Extension"
] | [
"Definition:Field Extension/Degree",
"Size of Linearly Independent Subset is at Most Size of Finite Generator",
"Definition:Linear Combination",
"Definition:Polynomial Function/Ring",
"Definition:Coefficient of Polynomial",
"Definition:Algebraic Element of Field Extension",
"Definition:Algebraic Field E... |
proofwiki-3719 | Finitely Generated Algebraic Extension is Finite | Let $L / K$ be a field extension.
Let $\alpha_1, \ldots, \alpha_n \in L$ be algebraic over $K$.
Let $K \sqbrk {\alpha_1, \ldots, \alpha_n}$ be the field generated by $\set {\alpha_1, \ldots, \alpha_n}$.
Then $K \sqbrk {\alpha_1, \ldots, \alpha_n} / K$ is a finite field extension. | Let $S = \set {\alpha_1, \ldots, \alpha_n}$.
We show by induction on $n$ that $K \sqbrk S / K$ is finite.
Clearly $K$ is finite over itself, so the result holds when $n = 0$.
Now suppose that for all sets $T \subseteq L$ with $\card T \le n - 1$ and each element of $T$ algebraic over $K$, $K \sqbrk T / K$ is finite.
We... | Let $L / K$ be a [[Definition:Field Extension|field extension]].
Let $\alpha_1, \ldots, \alpha_n \in L$ be [[Definition:Algebraic Element of Field Extension|algebraic]] over $K$.
Let $K \sqbrk {\alpha_1, \ldots, \alpha_n}$ be the field [[Definition:Generated Field Extension|generated]] by $\set {\alpha_1, \ldots, \al... | Let $S = \set {\alpha_1, \ldots, \alpha_n}$.
We show by [[Principle of Mathematical Induction|induction]] on $n$ that $K \sqbrk S / K$ is [[Definition:Finite Field Extension|finite]].
Clearly $K$ is finite over itself, so the result holds when $n = 0$.
Now suppose that for all sets $T \subseteq L$ with $\card T \le ... | Finitely Generated Algebraic Extension is Finite | https://proofwiki.org/wiki/Finitely_Generated_Algebraic_Extension_is_Finite | https://proofwiki.org/wiki/Finitely_Generated_Algebraic_Extension_is_Finite | [
"Field Extensions",
"Proofs by Induction"
] | [
"Definition:Field Extension",
"Definition:Algebraic Element of Field Extension",
"Definition:Generated Field Extension",
"Definition:Field Extension/Degree/Finite"
] | [
"Principle of Mathematical Induction",
"Definition:Field Extension/Degree/Finite",
"Definition:Algebraic Element of Field Extension",
"Finitely Generated Algebraic Extension is Finite",
"Structure of Simple Algebraic Field Extension",
"Degree Equation",
"Category:Field Extensions",
"Category:Proofs by... |
proofwiki-3720 | Transitivity of Algebraic Extensions | Let $E / F / K$ be a tower of field extensions.
Let $E$ be algebraic over $F$.
Let $F$ be algebraic over $K$.
Then $E$ is algebraic over $K$. | Let $x \in E$.
There are $a_0, \ldots, a_n \in F$ such that $a_0 + \cdots + a_n x^n = 0$.
Let $L = \map K {a_0, \ldots, a_n}$.
We have that $L / K$ is finitely generated and algebraic.
Therefore by Finitely Generated Algebraic Extension is Finite this extension is finite.
We have that $\map L x / L$ is simple and algeb... | Let $E / F / K$ be a [[Definition:Tower of Fields|tower]] of [[Definition:Field Extension|field extensions]].
Let $E$ be [[Definition:Algebraic Field Extension|algebraic]] over $F$.
Let $F$ be [[Definition:Algebraic Field Extension|algebraic]] over $K$.
Then $E$ is [[Definition:Algebraic Field Extension|algebraic]]... | Let $x \in E$.
There are $a_0, \ldots, a_n \in F$ such that $a_0 + \cdots + a_n x^n = 0$.
Let $L = \map K {a_0, \ldots, a_n}$.
We have that $L / K$ is [[Definition:Finitely Generated Field Extension|finitely generated]] and [[Definition:Algebraic Field Extension|algebraic]].
Therefore by [[Finitely Generated Algebr... | Transitivity of Algebraic Extensions | https://proofwiki.org/wiki/Transitivity_of_Algebraic_Extensions | https://proofwiki.org/wiki/Transitivity_of_Algebraic_Extensions | [
"Field Extensions"
] | [
"Definition:Tower of Fields",
"Definition:Field Extension",
"Definition:Algebraic Field Extension",
"Definition:Algebraic Field Extension",
"Definition:Algebraic Field Extension"
] | [
"Definition:Finitely Generated Field Extension",
"Definition:Algebraic Field Extension",
"Finitely Generated Algebraic Extension is Finite",
"Definition:Field Extension",
"Definition:Field Extension/Degree/Finite",
"Definition:Simple Field Extension",
"Definition:Algebraic Field Extension",
"Structure... |
proofwiki-3721 | Maximal Algebraic Extension is Subfield | Let $L / K$ be a field extension.
Let $K^a$ be the maximal algebraic extension of $K$ contained in $L$.
Then $K^a$ is a subfield of $L$. | Let $\alpha, \beta \in K^a$.
By Field Adjoined Algebraic Elements is Algebraic, $\map K {\alpha, \beta} / K$ is algebraic.
By definition, $\map K {\alpha, \beta}$ is a field.
Therefore $\alpha \beta$, $\alpha^{-1}$ and $\alpha - \beta$ all lie in $\map K {\alpha, \beta}$.
Hence all are algebraic over $K$.
Also:
$K \sub... | Let $L / K$ be a [[Definition:Field Extension|field extension]].
Let $K^a$ be the [[Definition:Maximal Algebraic Extension|maximal algebraic extension]] of $K$ contained in $L$.
Then $K^a$ is a [[Definition:Subfield|subfield]] of $L$. | Let $\alpha, \beta \in K^a$.
By [[Field Adjoined Algebraic Elements is Algebraic]], $\map K {\alpha, \beta} / K$ is [[Definition:Algebraic Field Extension|algebraic]].
By definition, $\map K {\alpha, \beta}$ is a [[Definition:Field (Abstract Algebra)|field]].
Therefore $\alpha \beta$, $\alpha^{-1}$ and $\alpha - \be... | Maximal Algebraic Extension is Subfield | https://proofwiki.org/wiki/Maximal_Algebraic_Extension_is_Subfield | https://proofwiki.org/wiki/Maximal_Algebraic_Extension_is_Subfield | [
"Field Extensions"
] | [
"Definition:Field Extension",
"Definition:Relative Algebraic Closure",
"Definition:Subfield"
] | [
"Field Adjoined Algebraic Elements is Algebraic",
"Definition:Algebraic Field Extension",
"Definition:Field (Abstract Algebra)",
"Definition:Algebraic Element of Field Extension",
"Subfield Test",
"Definition:Subfield",
"Category:Field Extensions"
] |
proofwiki-3722 | Equivalence of Definitions of Algebraically Closed Field | Let $K$ be a field.
{{TFAE|def = Algebraically Closed Field}} | === Definition $(1)$ implies Definition $(2)$ ===
Let $K$ be algebraically closed by definition 1.
Let $f$ be an irreducible polynomial over $K$.
By Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal, the ideal $\gen f$ generated by $f$ is maximal.
So by Maximal Ideal iff Quotient Ring is F... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
{{TFAE|def = Algebraically Closed Field}} | === Definition $(1)$ implies Definition $(2)$ ===
Let $K$ be [[Definition:Algebraically Closed Field/Definition 1|algebraically closed by definition 1]].
Let $f$ be an [[Definition:Irreducible Polynomial|irreducible polynomial]] over $K$.
By [[Principal Ideal of Principal Ideal Domain is of Irreducible Element iff M... | Equivalence of Definitions of Algebraically Closed Field | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Algebraically_Closed_Field | https://proofwiki.org/wiki/Equivalence_of_Definitions_of_Algebraically_Closed_Field | [
"Field Extensions"
] | [
"Definition:Field (Abstract Algebra)"
] | [
"Definition:Algebraically Closed Field/Definition 1",
"Definition:Irreducible Polynomial",
"Principal Ideal of Principal Ideal Domain is of Irreducible Element iff Maximal",
"Definition:Ideal of Ring",
"Definition:Generator of Ideal of Ring",
"Definition:Maximal Ideal of Ring",
"Maximal Ideal iff Quotie... |
proofwiki-3723 | Closed Set in Particular Point Space has no Limit Points | Let $T = \struct {S, \tau_p}$ be a particular point space.
Let $H \subsetneq S$ be closed in $T$.
Then $H$ has no limit points. | Let $H$ be closed in $T$.
Then by definition $p \notin H$.
Let $x \in H$.
By definition, $x$ is a limit point of $H$ if every open set $U \in \tau$ such that $x \in U$ contains some point of $H$ other than $x$.
Consider the set $U_x := \set {x, p} \subseteq S$.
As $p \in U_x$ we have that $U_x$ is open in $T$.
But ther... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Let $H \subsetneq S$ be [[Definition:Closed Set (Topology)|closed]] in $T$.
Then $H$ has no [[Definition:Limit Point of Set|limit points]]. | Let $H$ be [[Definition:Closed Set (Topology)|closed]] in $T$.
Then by definition $p \notin H$.
Let $x \in H$.
By definition, $x$ is a [[Definition:Limit Point of Set|limit point of $H$]] if every [[Definition:Open Set (Topology)|open set]] $U \in \tau$ such that $x \in U$ contains some point of $H$ other than $x$.... | Closed Set in Particular Point Space has no Limit Points | https://proofwiki.org/wiki/Closed_Set_in_Particular_Point_Space_has_no_Limit_Points | https://proofwiki.org/wiki/Closed_Set_in_Particular_Point_Space_has_no_Limit_Points | [
"Particular Point Topologies",
"Examples of Limit Points",
"Closed Sets"
] | [
"Definition:Particular Point Topology",
"Definition:Closed Set/Topology",
"Definition:Limit Point/Topology/Set"
] | [
"Definition:Closed Set/Topology",
"Definition:Limit Point/Topology/Set",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Limit Point/Topology/Set",
"Definition:Limit Point/Topology/Set"
] |
proofwiki-3724 | Particular Point Space is Scattered | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is scattered. | {{Recall|Scattered Space|scattered space}}
{{:Definition:Scattered Space/Definition 1}}
Let $H \subseteq S$ such that $p \notin H$.
From Subset of Particular Point Space is either Open or Closed, $H$ is closed in $T$.
We have that Closed Set in Particular Point Space has no Limit Points.
So if $p \notin H$ then $H$ has... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Space|particular point space]].
Then $T$ is [[Definition:Scattered Space|scattered]]. | {{Recall|Scattered Space|scattered space}}
{{:Definition:Scattered Space/Definition 1}}
Let $H \subseteq S$ such that $p \notin H$.
From [[Subset of Particular Point Space is either Open or Closed]], $H$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
We have that [[Closed Set in Particular Point Space has no... | Particular Point Space is Scattered | https://proofwiki.org/wiki/Particular_Point_Space_is_Scattered | https://proofwiki.org/wiki/Particular_Point_Space_is_Scattered | [
"Particular Point Topologies",
"Examples of Scattered Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Scattered Space"
] | [
"Subset of Particular Point Space is either Open or Closed",
"Definition:Closed Set/Topology",
"Closed Set in Particular Point Space has no Limit Points",
"Definition:Limit Point/Topology/Set",
"Definition:Element",
"Definition:Set",
"Definition:Isolated Point (Topology)/Subset",
"Definition:Dense-in-... |
proofwiki-3725 | Cook-Levin Theorem | The boolean satisfiability problem is NP-Complete. | === The Boolean Satisfiability Problem is NP ===
Given a boolean satisfiability problem with a set of variables $X$ and clauses $L$ and a possible solution to the problem, it is a trivial matter to evaluate all the clauses in $L$ to verify the solution in polynomial time.
{{Handwaving|The above needs to be justified wi... | The [[Definition:Boolean Satisfiability Problem|boolean satisfiability problem]] is [[Definition:NP-Complete|NP-Complete]]. | === The Boolean Satisfiability Problem is NP ===
Given a [[Definition:Boolean Satisfiability Problem|boolean satisfiability problem]] with a set of variables $X$ and clauses $L$ and a possible solution to the problem, it is a trivial matter to evaluate all the clauses in $L$ to verify the solution in polynomial time.
... | Cook-Levin Theorem | https://proofwiki.org/wiki/Cook-Levin_Theorem | https://proofwiki.org/wiki/Cook-Levin_Theorem | [
"Boolean Satisfiability Problems",
"Complexity Theory"
] | [
"Definition:Boolean Satisfiability Problem",
"Definition:NP-Complete"
] | [
"Definition:Boolean Satisfiability Problem",
"NP Problem iff Solution Verifiable in Polynomial Time",
"Definition:NP Complexity Class",
"Definition:Boolean Satisfiability Problem",
"Definition:NP Complexity Class",
"Definition:Boolean Satisfiability Problem",
"Definition:Boolean Satisfiability Problem",... |
proofwiki-3726 | Particular Point Space is Irreducible | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is irreducible. | By definition, $T = \struct {S, \tau_p}$ is irreducible {{iff}} every two non-empty open sets of $T$ have non-empty intersection.
Let $U_1$ and $U_2$ be non-empty open sets of $T$.
By definition of particular point space, $p \in U_1$ and $p \in U_2$.
Thus:
:$p \in U_1 \cap U_2$
and so:
:$U_1 \cap U_2 \ne \O$
Hence the ... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $T$ is [[Definition:Irreducible Space|irreducible]]. | By definition, $T = \struct {S, \tau_p}$ is [[Definition:Irreducible Space|irreducible]] {{iff}} every two [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Set (Topology)|open sets]] of $T$ have [[Definition:Non-Empty Set|non-empty]] [[Definition:Set Intersection|intersection]].
Let $U_1$ and $U_2$ be [[Defini... | Particular Point Space is Irreducible/Proof 1 | https://proofwiki.org/wiki/Particular_Point_Space_is_Irreducible | https://proofwiki.org/wiki/Particular_Point_Space_is_Irreducible/Proof_1 | [
"Particular Point Space is Irreducible",
"Particular Point Topologies",
"Examples of Irreducible Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Irreducible Space"
] | [
"Definition:Irreducible Space",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Definition:Non-Empty Set",
"Definition:Set Intersection",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Definition:Particular Point Topology"
] |
proofwiki-3727 | Particular Point Space is not Ultraconnected | Let $T = \struct {S, \tau_p}$ be a particular point space with at least three points.
Then $T$ is not ultraconnected. | Let $x, y \in S: x \ne p, y \ne p, x \ne y$.
Consider $\set x$ and $\set y$.
Neither are open as neither contain $p$.
So from Subset of Particular Point Space is either Open or Closed they are both closed.
We have that $\set x \cap \set y = \O$.
The result follows by definition of ultraconnected.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]] with at least three [[Definition:Point of Set|points]].
Then $T$ is not [[Definition:Ultraconnected Space|ultraconnected]]. | Let $x, y \in S: x \ne p, y \ne p, x \ne y$.
Consider $\set x$ and $\set y$.
Neither are [[Definition:Open Set (Topology)|open]] as neither contain $p$.
So from [[Subset of Particular Point Space is either Open or Closed]] they are both [[Definition:Closed Set (Topology)|closed]].
We have that $\set x \cap \set y =... | Particular Point Space is not Ultraconnected | https://proofwiki.org/wiki/Particular_Point_Space_is_not_Ultraconnected | https://proofwiki.org/wiki/Particular_Point_Space_is_not_Ultraconnected | [
"Particular Point Topologies",
"Examples of Ultraconnected Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Element",
"Definition:Ultraconnected Space"
] | [
"Definition:Open Set/Topology",
"Subset of Particular Point Space is either Open or Closed",
"Definition:Closed Set/Topology",
"Definition:Ultraconnected Space"
] |
proofwiki-3728 | Dispersion Point in Particular Point Space | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $p$ is dispersion point of $T$. | Let $H = S \setminus \set p$.
Let $T_H = \struct {H, \tau_H}$ be the topological subspace induced on $H$ by $\tau_p$.
From Particular Point Space less Particular Point is Discrete, the space $T_H$ is discrete.
We have Discrete Space is Locally Connected.
Thus from Totally Disconnected and Locally Connected Space is Dis... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $p$ is [[Definition:Dispersion Point|dispersion point]] of $T$. | Let $H = S \setminus \set p$.
Let $T_H = \struct {H, \tau_H}$ be the [[Definition:Topological Subspace|topological subspace induced on $H$ by $\tau_p$]].
From [[Particular Point Space less Particular Point is Discrete]], the space $T_H$ is [[Definition:Discrete Space|discrete]].
We have [[Discrete Space is Locally ... | Dispersion Point in Particular Point Space | https://proofwiki.org/wiki/Dispersion_Point_in_Particular_Point_Space | https://proofwiki.org/wiki/Dispersion_Point_in_Particular_Point_Space | [
"Particular Point Topologies",
"Examples of Dispersion Points"
] | [
"Definition:Particular Point Topology",
"Definition:Dispersion Point"
] | [
"Definition:Topological Subspace",
"Particular Point Space less Particular Point is Discrete",
"Definition:Discrete Topology",
"Discrete Space is Locally Connected",
"Totally Disconnected and Locally Connected Space is Discrete",
"Definition:Totally Disconnected Space",
"Definition:Dispersion Point"
] |
proofwiki-3729 | Infinite Particular Point Space is not Weakly Countably Compact | Let $T = \struct {S, \tau_p}$ be an infinite particular point space.
Then $T$ is not weakly countably compact. | {{Recall|Weakly Countably Compact Space|weakly countably compact space}}
{{:Definition:Weakly Countably Compact Space}}
So, let $T = \struct {S, \tau_p}$ be an infinite particular point space {{WRT}} the particular point $p$.
Let $H \subseteq S$ be an infinite subset of $S$ where $p \notin H$.
$H$ is not open in $T$ by... | Let $T = \struct {S, \tau_p}$ be an [[Definition:Infinite Particular Point Space|infinite particular point space]].
Then $T$ is not [[Definition:Weakly Countably Compact Space|weakly countably compact]]. | {{Recall|Weakly Countably Compact Space|weakly countably compact space}}
{{:Definition:Weakly Countably Compact Space}}
So, let $T = \struct {S, \tau_p}$ be an [[Definition:Infinite Particular Point Topology|infinite particular point space]] {{WRT}} the [[Definition:Particular Point|particular point]] $p$.
Let $H \su... | Infinite Particular Point Space is not Weakly Countably Compact | https://proofwiki.org/wiki/Infinite_Particular_Point_Space_is_not_Weakly_Countably_Compact | https://proofwiki.org/wiki/Infinite_Particular_Point_Space_is_not_Weakly_Countably_Compact | [
"Infinite Particular Point Topologies",
"Examples of Weakly Countably Compact Spaces"
] | [
"Definition:Particular Point Topology/Infinite",
"Definition:Weakly Countably Compact Space"
] | [
"Definition:Particular Point Topology/Infinite",
"Definition:Particular Point",
"Definition:Infinite Set",
"Definition:Subset",
"Definition:Open Set/Topology",
"Subset of Particular Point Space is either Open or Closed",
"Definition:Closed Set/Topology",
"Closed Set in Particular Point Space has no Li... |
proofwiki-3730 | Fundamental Theorem of Symmetric Polynomials | Let $K$ be a field.
Let $f$ a symmetric polynomial over $K$.
Then $f$ can be written uniquely as a polynomial in the elementary symmetric polynomials. | Let $p \in R := K \sqbrk {x_1, \dotsc, x_n}$ be a symmetric polynomial.
Let $\sigma_1, \dotsc, \sigma_n$ denote the $n$ elementary symmetric polynomials.
Let $<$ be the degree reverse lexicographic monomial order.
Let us denote by $\map {LM} f$ the leading monomial of any $f \in R$, that is to say the unique $<$-larges... | Let $K$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $f$ a [[Definition:Symmetric Polynomial|symmetric polynomial]] over $K$.
Then $f$ can be written uniquely as a [[Definition:Polynomial|polynomial]] in the [[Definition:Elementary Symmetric Polynomial|elementary symmetric polynomials]]. | Let $p \in R := K \sqbrk {x_1, \dotsc, x_n}$ be a [[Definition:Symmetric Polynomial|symmetric polynomial]].
Let $\sigma_1, \dotsc, \sigma_n$ denote the $n$ [[Definition:Elementary Symmetric Polynomial|elementary symmetric polynomials]].
Let $<$ be the [[Definition:Degree Reverse Lexicographic Monomial Order|degree re... | Fundamental Theorem of Symmetric Polynomials | https://proofwiki.org/wiki/Fundamental_Theorem_of_Symmetric_Polynomials | https://proofwiki.org/wiki/Fundamental_Theorem_of_Symmetric_Polynomials | [
"Fundamental Theorem of Symmetric Polynomials",
"Elementary Symmetric Polynomials",
"Symmetric Polynomials",
"Fundamental Theorems"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Symmetric Polynomial",
"Definition:Polynomial",
"Definition:Elementary Symmetric Polynomial"
] | [
"Definition:Symmetric Polynomial",
"Definition:Elementary Symmetric Polynomial",
"Definition:Degree Reverse Lexicographic Monomial Order",
"Definition:Leading Monomial",
"Definition:Monomial",
"Definition:Monomial",
"Definition:Monomial",
"Definition:Polynomial",
"Definition:Symmetric Polynomial",
... |
proofwiki-3731 | Particular Point Space is Pseudocompact | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is a pseudocompact space. | We have that:
:a Particular Point Space is Irreducible
:an Irreducible Space is Pseudocompact.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $T$ is a [[Definition:Pseudocompact Space|pseudocompact space]]. | We have that:
:a [[Particular Point Space is Irreducible]]
:an [[Irreducible Space is Pseudocompact]].
{{qed}} | Particular Point Space is Pseudocompact | https://proofwiki.org/wiki/Particular_Point_Space_is_Pseudocompact | https://proofwiki.org/wiki/Particular_Point_Space_is_Pseudocompact | [
"Particular Point Topologies",
"Examples of Pseudocompact Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Pseudocompact Space"
] | [
"Particular Point Space is Irreducible",
"Irreducible Space is Pseudocompact"
] |
proofwiki-3732 | Particular Point Space is Path-Connected | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is path-connected. | Let $q \in S$.
Let $\mathbb I$ be the closed unit interval in $\R$.
Let $f: \mathbb I \to S$ be the mapping defined as:
:<nowiki>$\forall x \in \mathbb I: \map f x = \begin{cases}
p & : x \in \hointr 0 1 \\
q & : x = 1
\end{cases}$</nowiki>
Let $U \in \tau_p$.
Then by definition of particular point space, $p \in U$
Eit... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $T$ is [[Definition:Path-Connected Space|path-connected]]. | Let $q \in S$.
Let $\mathbb I$ be the [[Definition:Closed Unit Interval|closed unit interval]] in $\R$.
Let $f: \mathbb I \to S$ be the [[Definition:Mapping|mapping]] defined as:
:<nowiki>$\forall x \in \mathbb I: \map f x = \begin{cases}
p & : x \in \hointr 0 1 \\
q & : x = 1
\end{cases}$</nowiki>
Let $U \in \tau_... | Particular Point Space is Path-Connected | https://proofwiki.org/wiki/Particular_Point_Space_is_Path-Connected | https://proofwiki.org/wiki/Particular_Point_Space_is_Path-Connected | [
"Particular Point Topologies",
"Examples of Path-Connected Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Path-Connected/Topological Space"
] | [
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Mapping",
"Definition:Particular Point Topology",
"Definition:Open Set/Topology",
"Definition:Real Interval/Half-Open",
"Definition:Open Set/Topology",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Path (Topology)",
"P... |
proofwiki-3733 | Particular Point Space is not Injectively Path-Connected | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is not injectively path-connected. | Let $q \in S$ be such that $q \ne p$.
Let $f: \closedint 0 1 \to T$ be an injection such that:
{{begin-eqn}}
{{eqn | l = \map f 0
| r = q
}}
{{eqn | l = \map f 1
| r = p
}}
{{end-eqn}}
Because $f$ is an injection, it must be that:
:$\map {f^{-1} } {\set p} = \set 1$
where $f^{-1}$ denotes the preimage under... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $T$ is not [[Definition:Injectively Path-Connected Space|injectively path-connected]]. | Let $q \in S$ be such that $q \ne p$.
Let $f: \closedint 0 1 \to T$ be an [[Definition:Injection|injection]] such that:
{{begin-eqn}}
{{eqn | l = \map f 0
| r = q
}}
{{eqn | l = \map f 1
| r = p
}}
{{end-eqn}}
Because $f$ is an [[Definition:Injection|injection]], it must be that:
:$\map {f^{-1} } {\set ... | Particular Point Space is not Injectively Path-Connected | https://proofwiki.org/wiki/Particular_Point_Space_is_not_Injectively_Path-Connected | https://proofwiki.org/wiki/Particular_Point_Space_is_not_Injectively_Path-Connected | [
"Particular Point Topologies",
"Examples of Injectively Path-Connected Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Injectively Path-Connected/Topological Space"
] | [
"Definition:Injection",
"Definition:Injection",
"Definition:Preimage/Mapping/Subset",
"Definition:Open Set/Topology",
"Definition:Particular Point Topology",
"Closed Real Interval is not Open Set",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Preimage/Mapping/Subset",
... |
proofwiki-3734 | Basis for Particular Point Space | Let $T = \struct {S, \tau_p}$ be a particular point space.
Consider the set $\BB$ defined as:
:$\BB = \set {\set {x, p}: x \in S} \cup \set p$
Then $B$ is a basis for $S$. | Let $H \in \tau_p$ be open in $T$.
Then:
:$\forall y \in H: \exists \set {y, p} \in \BB$
which also holds when $y = p$ as $\set {y, p} = \set p \in \BB$.
Thus:
:$\ds H = \bigcup_{y \mathop \in H} \set {y, p}$
So $\BB$ is an analytic basis for $T$.
{{qed}}
It could equally well be shown that $\BB$ is also a synthetic ba... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Consider the set $\BB$ defined as:
:$\BB = \set {\set {x, p}: x \in S} \cup \set p$
Then $B$ is a [[Definition:Basis (Topology)|basis]] for $S$. | Let $H \in \tau_p$ be [[Definition:Open Set (Topology)|open]] in $T$.
Then:
:$\forall y \in H: \exists \set {y, p} \in \BB$
which also holds when $y = p$ as $\set {y, p} = \set p \in \BB$.
Thus:
:$\ds H = \bigcup_{y \mathop \in H} \set {y, p}$
So $\BB$ is an [[Definition:Analytic Basis|analytic basis]] for $T$.
{{qe... | Basis for Particular Point Space | https://proofwiki.org/wiki/Basis_for_Particular_Point_Space | https://proofwiki.org/wiki/Basis_for_Particular_Point_Space | [
"Particular Point Topologies",
"Topological Bases"
] | [
"Definition:Particular Point Topology",
"Definition:Basis (Topology)"
] | [
"Definition:Open Set/Topology",
"Definition:Basis (Topology)/Analytic Basis",
"Definition:Basis (Topology)/Synthetic Basis",
"Category:Particular Point Topologies",
"Category:Topological Bases"
] |
proofwiki-3735 | Particular Point Space is Locally Path-Connected | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is locally path-connected. | Consider the set $\BB$ defined as:
:$\BB = \set {\set {x, p}: x \in S}$
Then $\BB$ is a basis for $T$.
Now consider the open set $\set {p, q} \in \BB$.
Let $\mathbb I$ be the closed unit interval in $\R$.
Let $f: \mathbb I \to S$ be the mapping defined as:
:<nowiki>$\forall x \in \mathbb I: \map f x = \begin{cases}
p &... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $T$ is [[Definition:Locally Path-Connected Space|locally path-connected]]. | Consider the [[Definition:Set|set]] $\BB$ defined as:
:$\BB = \set {\set {x, p}: x \in S}$
Then $\BB$ is a [[Basis for Particular Point Space|basis for $T$]].
Now consider the [[Definition:Open Set (Topology)|open set]] $\set {p, q} \in \BB$.
Let $\mathbb I$ be the [[Definition:Closed Unit Interval|closed unit int... | Particular Point Space is Locally Path-Connected | https://proofwiki.org/wiki/Particular_Point_Space_is_Locally_Path-Connected | https://proofwiki.org/wiki/Particular_Point_Space_is_Locally_Path-Connected | [
"Particular Point Topologies",
"Examples of Locally Path-Connected Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Locally Path-Connected Space"
] | [
"Definition:Set",
"Basis for Particular Point Space",
"Definition:Open Set/Topology",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Mapping",
"Definition:Open Set/Topology",
"Definition:Real Interval/Half-Open",
"Definition:Open Set/Topology",
"Definition:Continuous Mapping (Topology)... |
proofwiki-3736 | Particular Point Space is Non-Meager | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is non-meager. | Suppose $T$ were meager.
Then it would be a countable union of subsets which are nowhere dense in $T$.
Let $H \subseteq S$.
From Closure of Open Set of Particular Point Space, the closure of $H$ is $S$.
From the definition of interior, the interior of $S$ is $S$.
So the interior of the closure of $H$ is not empty.
So $... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $T$ is [[Definition:Non-Meager Space|non-meager]]. | Suppose $T$ were [[Definition:Meager Space|meager]].
Then it would be a [[Definition:Countable Union|countable union]] of [[Definition:Subset|subsets]] which are [[Definition:Nowhere Dense|nowhere dense]] in $T$.
Let $H \subseteq S$.
From [[Closure of Open Set of Particular Point Space]], the [[Definition:Closure (... | Particular Point Space is Non-Meager/Proof 1 | https://proofwiki.org/wiki/Particular_Point_Space_is_Non-Meager | https://proofwiki.org/wiki/Particular_Point_Space_is_Non-Meager/Proof_1 | [
"Particular Point Space is Non-Meager",
"Particular Point Topologies",
"Examples of Non-Meager Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Meager Space/Non-Meager"
] | [
"Definition:Meager Space",
"Definition:Set Union/Countable Union",
"Definition:Subset",
"Definition:Nowhere Dense",
"Closure of Open Set of Particular Point Space",
"Definition:Closure (Topology)",
"Definition:Interior (Topology)",
"Definition:Interior (Topology)",
"Definition:Interior (Topology)",
... |
proofwiki-3737 | Particular Point Space is Non-Meager | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is non-meager. | By definition of particular point space, any subset of $S$ which contains $p$ is open in $T$.
So $\left\{{p}\right\}$ itself is open in $T$.
That is, $p$ is an open point.
The result follows from Space with Open Point is Non-Meager.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $T$ is [[Definition:Non-Meager Space|non-meager]]. | By definition of [[Definition:Particular Point Topology|particular point space]], any [[Definition:Subset|subset]] of $S$ which contains $p$ is [[Definition:Open Set (Topology)|open]] in $T$.
So $\left\{{p}\right\}$ itself is [[Definition:Open Set (Topology)|open]] in $T$.
That is, $p$ is an [[Definition:Open Point|o... | Particular Point Space is Non-Meager/Proof 2 | https://proofwiki.org/wiki/Particular_Point_Space_is_Non-Meager | https://proofwiki.org/wiki/Particular_Point_Space_is_Non-Meager/Proof_2 | [
"Particular Point Space is Non-Meager",
"Particular Point Topologies",
"Examples of Non-Meager Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Meager Space/Non-Meager"
] | [
"Definition:Particular Point Topology",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Point",
"Space with Open Point is Non-Meager"
] |
proofwiki-3738 | Particular Point Space is Non-Meager | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is non-meager. | By definition of particular point space, any subset of $S$ which contains $p$ is open in $T$.
{{AimForCont}} $T$ is meager.
By definition, $T$ is meager {{iff}} it is a countable union of subsets of $S$ which are nowhere dense in $T$.
At least one such nowhere dense subset $U$ of $S$ must contain $p$.
By definition, $U... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Then $T$ is [[Definition:Non-Meager Space|non-meager]]. | By definition of [[Definition:Particular Point Topology|particular point space]], any [[Definition:Subset|subset]] of $S$ which contains $p$ is [[Definition:Open Set (Topology)|open]] in $T$.
{{AimForCont}} $T$ is [[Definition:Meager Space|meager]].
By definition, $T$ is [[Definition:Meager Space|meager]] {{iff}} it... | Particular Point Space is Non-Meager/Proof 3 | https://proofwiki.org/wiki/Particular_Point_Space_is_Non-Meager | https://proofwiki.org/wiki/Particular_Point_Space_is_Non-Meager/Proof_3 | [
"Particular Point Space is Non-Meager",
"Particular Point Topologies",
"Examples of Non-Meager Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Meager Space/Non-Meager"
] | [
"Definition:Particular Point Topology",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Meager Space",
"Definition:Meager Space",
"Definition:Set Union/Countable Union",
"Definition:Subset",
"Definition:Nowhere Dense",
"Definition:Nowhere Dense",
"Definition:Subset",
"Definition... |
proofwiki-3739 | Infinite Particular Point Space is not Countably Metacompact | Let $T = \struct {S, \tau_p}$ be an infinite particular point space.
Then $T$ is not countably metacompact. | {{Recall|Countably Metacompact Space|countably metacompact}}
{{:Definition:Countably Metacompact Space}}
;Countable case
Suppose $T$ is a countable particular point space.
Let $\CC$ be the open cover of $T$ defined as:
:$\CC = \set {\set {x, p}: x \in S}$
$\CC$ is countable and has no open refinement except $\CC$ itsel... | Let $T = \struct {S, \tau_p}$ be an [[Definition:Infinite Particular Point Topology|infinite particular point space]].
Then $T$ is not [[Definition:Countably Metacompact Space|countably metacompact]]. | {{Recall|Countably Metacompact Space|countably metacompact}}
{{:Definition:Countably Metacompact Space}}
;[[Definition:Countable Particular Point Topology|Countable case]]
Suppose $T$ is a [[Definition:Countable Particular Point Topology|countable particular point space]].
Let $\CC$ be the [[Definition:Open Cover|op... | Infinite Particular Point Space is not Countably Metacompact | https://proofwiki.org/wiki/Infinite_Particular_Point_Space_is_not_Countably_Metacompact | https://proofwiki.org/wiki/Infinite_Particular_Point_Space_is_not_Countably_Metacompact | [
"Infinite Particular Point Topologies",
"Examples of Countably Metacompact Spaces"
] | [
"Definition:Particular Point Topology/Infinite",
"Definition:Countably Metacompact Space"
] | [
"Definition:Particular Point Topology/Countable",
"Definition:Particular Point Topology/Countable",
"Definition:Open Cover",
"Definition:Countable Set",
"Definition:Open Refinement",
"Definition:Point Finite Cover",
"Definition:Countably Infinite/Set",
"Definition:Particular Point Topology/Uncountable... |
proofwiki-3740 | Exportation and Self-Conditional | :$p \land q \implies r \dashv \vdash \paren {p \implies q} \implies \paren {p \implies r}$ | From the Rule of Exportation:
:$\paren {p \land q} \implies r \dashv \vdash p \implies \paren {q \implies r}$
Then by Self-Distributive Law for Conditional:
:$p \implies \paren {q \implies r} \dashv \vdash \paren {p \implies q} \implies \paren {p \implies r}$
{{qed}}
Category:Conditional
Category:Conjunction
mgps1nmi22... | :$p \land q \implies r \dashv \vdash \paren {p \implies q} \implies \paren {p \implies r}$ | From the [[Rule of Exportation]]:
:$\paren {p \land q} \implies r \dashv \vdash p \implies \paren {q \implies r}$
Then by [[Self-Distributive Law for Conditional]]:
:$p \implies \paren {q \implies r} \dashv \vdash \paren {p \implies q} \implies \paren {p \implies r}$
{{qed}}
[[Category:Conditional]]
[[Category:Conj... | Exportation and Self-Conditional | https://proofwiki.org/wiki/Exportation_and_Self-Conditional | https://proofwiki.org/wiki/Exportation_and_Self-Conditional | [
"Conditional",
"Conjunction"
] | [] | [
"Rule of Exportation",
"Self-Distributive Law for Conditional",
"Category:Conditional",
"Category:Conjunction"
] |
proofwiki-3741 | Łoś-Vaught Test | Let $T$ be a satisfiable $\LL$-theory with no finite models.
Let $T$ be $\kappa$-categorical for some infinite cardinal $\kappa \ge \card \LL$.
Then $T$ is complete. | We prove the contrapositive.
The main idea is that if such a theory $T$ is incomplete, we can construct size $\kappa$ models which disagree on a sentence.
Suppose $T$ is not complete.
By the definition of complete, this means that there is some sentence $\phi$ such that both $T \not \models \phi$ and $T \not \models \n... | Let $T$ be a satisfiable $\LL$-theory with no finite models.
Let $T$ be $\kappa$-[[Definition:Categorical (Model Theory)|categorical]] for some infinite cardinal $\kappa \ge \card \LL$.
Then $T$ is complete. | We prove the contrapositive.
The main idea is that if such a theory $T$ is incomplete, we can construct size $\kappa$ models which disagree on a sentence.
Suppose $T$ is not complete.
By the definition of complete, this means that there is some sentence $\phi$ such that both $T \not \models \phi$ and $T \not \model... | Łoś-Vaught Test | https://proofwiki.org/wiki/Łoś-Vaught_Test | https://proofwiki.org/wiki/Łoś-Vaught_Test | [
"Model Theory for Predicate Logic"
] | [
"Definition:Categorical (Model Theory)"
] | [
"Upward Löwenheim-Skolem Theorem",
"Łoś-Vaught Test",
"Category:Model Theory for Predicate Logic"
] |
proofwiki-3742 | Sierpiński Space is Irreducible | Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.
Then $T$ is irreducible. | A Sierpiński space is a particular point space by definition.
A Particular Point Space is Irreducible.
{{qed}} | Let $T = \struct {\set {0, 1}, \tau_0}$ be a [[Definition:Sierpiński Space|Sierpiński space]].
Then $T$ is [[Definition:Irreducible Space|irreducible]]. | A [[Definition:Sierpiński Space|Sierpiński space]] is a [[Definition:Particular Point Space|particular point space]] by definition.
A [[Particular Point Space is Irreducible]].
{{qed}} | Sierpiński Space is Irreducible | https://proofwiki.org/wiki/Sierpiński_Space_is_Irreducible | https://proofwiki.org/wiki/Sierpiński_Space_is_Irreducible | [
"Sierpiński Space",
"Examples of Irreducible Spaces"
] | [
"Definition:Sierpiński Space",
"Definition:Irreducible Space"
] | [
"Definition:Sierpiński Space",
"Definition:Particular Point Topology",
"Particular Point Space is Irreducible"
] |
proofwiki-3743 | Sierpiński Space is Ultraconnected | Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.
Then $T$ is ultraconnected. | The only closed sets of $T$ are $\O, \set 1$ and $\set {0, 1}$.
$\set 1$ and $\set {0, 1}$ are not disjoint.
Hence the result by definition of ultraconnected.
{{qed}} | Let $T = \struct {\set {0, 1}, \tau_0}$ be a [[Definition:Sierpiński Space|Sierpiński space]].
Then $T$ is [[Definition:Ultraconnected Space|ultraconnected]]. | The only [[Definition:Closed Set (Topology)|closed sets]] of $T$ are $\O, \set 1$ and $\set {0, 1}$.
$\set 1$ and $\set {0, 1}$ are not [[Definition:Disjoint Sets|disjoint]].
Hence the result by definition of [[Definition:Ultraconnected Space|ultraconnected]].
{{qed}} | Sierpiński Space is Ultraconnected | https://proofwiki.org/wiki/Sierpiński_Space_is_Ultraconnected | https://proofwiki.org/wiki/Sierpiński_Space_is_Ultraconnected | [
"Sierpiński Space",
"Examples of Ultraconnected Spaces"
] | [
"Definition:Sierpiński Space",
"Definition:Ultraconnected Space"
] | [
"Definition:Closed Set/Topology",
"Definition:Disjoint Sets",
"Definition:Ultraconnected Space"
] |
proofwiki-3744 | Sierpiński Space is Path-Connected | Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.
Then $T$ is path-connected. | A Sierpiński space is a particular point space by definition.
A Particular Point Space is Path-Connected.
{{qed}} | Let $T = \struct {\set {0, 1}, \tau_0}$ be a [[Definition:Sierpiński Space|Sierpiński space]].
Then $T$ is [[Definition:Path-Connected Space|path-connected]]. | A [[Definition:Sierpiński Space|Sierpiński space]] is a [[Definition:Particular Point Space|particular point space]] by definition.
A [[Particular Point Space is Path-Connected]].
{{qed}} | Sierpiński Space is Path-Connected | https://proofwiki.org/wiki/Sierpiński_Space_is_Path-Connected | https://proofwiki.org/wiki/Sierpiński_Space_is_Path-Connected | [
"Sierpiński Space",
"Examples of Path-Connected Spaces"
] | [
"Definition:Sierpiński Space",
"Definition:Path-Connected/Topological Space"
] | [
"Definition:Sierpiński Space",
"Definition:Particular Point Topology",
"Particular Point Space is Path-Connected"
] |
proofwiki-3745 | Sierpiński Space is not Injectively Path-Connected | Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.
Then $T$ is not injectively path-connected. | A Sierpiński space is a particular point space by definition.
A Particular Point Space is not Injectively Path-Connected.
{{qed}} | Let $T = \struct {\set {0, 1}, \tau_0}$ be a [[Definition:Sierpiński Space|Sierpiński space]].
Then $T$ is not [[Definition:Injectively Path-Connected Space|injectively path-connected]]. | A [[Definition:Sierpiński Space|Sierpiński space]] is a [[Definition:Particular Point Space|particular point space]] by definition.
A [[Particular Point Space is not Injectively Path-Connected]].
{{qed}} | Sierpiński Space is not Injectively Path-Connected | https://proofwiki.org/wiki/Sierpiński_Space_is_not_Injectively_Path-Connected | https://proofwiki.org/wiki/Sierpiński_Space_is_not_Injectively_Path-Connected | [
"Sierpiński Space",
"Examples of Injectively Path-Connected Spaces"
] | [
"Definition:Sierpiński Space",
"Definition:Injectively Path-Connected/Topological Space"
] | [
"Definition:Sierpiński Space",
"Definition:Particular Point Topology",
"Particular Point Space is not Injectively Path-Connected"
] |
proofwiki-3746 | Sierpiński Space is T5 | Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.
Then $T$ is a $T_5$ space. | {{Recall|T5 Space|$T_5$ space}}
{{:Definition:T5 Space/Definition 1}}
The only closed sets in $T$ are $\O$, $\set 1$ and $\set {0, 1}$.
So there are no two separated sets $A, B \subseteq \set {0, 1}$.
So $T$ is a $T_5$ space vacuously.
{{qed}} | Let $T = \struct {\set {0, 1}, \tau_0}$ be a [[Definition:Sierpiński Space|Sierpiński space]].
Then $T$ is a [[Definition:T5 Space|$T_5$ space]]. | {{Recall|T5 Space|$T_5$ space}}
{{:Definition:T5 Space/Definition 1}}
The only [[Definition:Closed Set (Topology)|closed sets]] in $T$ are $\O$, $\set 1$ and $\set {0, 1}$.
So there are no two [[Definition:Separated Sets|separated sets]] $A, B \subseteq \set {0, 1}$.
So $T$ is a [[Definition:T5 Space|$T_5$ space]] v... | Sierpiński Space is T5 | https://proofwiki.org/wiki/Sierpiński_Space_is_T5 | https://proofwiki.org/wiki/Sierpiński_Space_is_T5 | [
"Sierpiński Space",
"Examples of T5 Spaces"
] | [
"Definition:Sierpiński Space",
"Definition:T5 Space"
] | [
"Definition:Closed Set/Topology",
"Definition:Separated Sets",
"Definition:T5 Space"
] |
proofwiki-3747 | Sierpiński Space is T4 | Let $T = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.
Then $T$ is a $T_4$ space. | We have that the Sierpiński Space is $T_5$.
Then we have that a $T_5$ Space is $T_4$.
{{qed}} | Let $T = \struct {\set {0, 1}, \tau_0}$ be a [[Definition:Sierpiński Space|Sierpiński space]].
Then $T$ is a [[Definition:T4 Space|$T_4$ space]]. | We have that the [[Sierpiński Space is T5|Sierpiński Space is $T_5$]].
Then we have that a [[T5 Space is T4|$T_5$ Space is $T_4$]].
{{qed}} | Sierpiński Space is T4 | https://proofwiki.org/wiki/Sierpiński_Space_is_T4 | https://proofwiki.org/wiki/Sierpiński_Space_is_T4 | [
"Sierpiński Space",
"Examples of T4 Spaces"
] | [
"Definition:Sierpiński Space",
"Definition:T4 Space"
] | [
"Sierpiński Space is T5",
"T5 Space is T4"
] |
proofwiki-3748 | Upward Löwenheim-Skolem Theorem | {{Disambiguate|Definition:Model|I suspect model of a first-order theory $\LL$, which is more specific than what is linked to now}}
Let $T$ be an $\LL$-theory with an infinite model.
Then for each infinite cardinal $\kappa \ge \card \LL$, there exists a model of $T$ with cardinality $\kappa$. | {{Explain|Missing reference to the use of Axiom of Dependent Choice or Axiom of Choice/Zorn's Lemma}}
The idea is:
:to extend the language by adding $\kappa$ many new constants
and:
:to extend the theory by adding sentences asserting that these constants are distinct.
It is shown that this new theory is finitely satis... | {{Disambiguate|Definition:Model|I suspect model of a first-order theory $\LL$, which is more specific than what is linked to now}}
Let $T$ be an $\LL$-theory with an infinite [[Definition:Model (Logic)|model]].
Then for each infinite cardinal $\kappa \ge \card \LL$, there exists a [[Definition:Model (Logic)|model]] of... | {{Explain|Missing reference to the use of Axiom of Dependent Choice or Axiom of Choice/Zorn's Lemma}}
The idea is:
:to extend the language by adding $\kappa$ many new constants
and:
:to extend the theory by adding sentences asserting that these constants are distinct.
It is shown that this new theory is finitely sa... | Upward Löwenheim-Skolem Theorem | https://proofwiki.org/wiki/Upward_Löwenheim-Skolem_Theorem | https://proofwiki.org/wiki/Upward_Löwenheim-Skolem_Theorem | [
"Model Theory for Predicate Logic"
] | [
"Definition:Model (Logic)",
"Definition:Model (Logic)"
] | [
"Compactness Theorem"
] |
proofwiki-3749 | Compactness Theorem | Let $\LL$ be the language of predicate logic.
Let $T$ be a set of $\LL$-sentences.
Then $T$ is satisfiable {{iff}} $T$ is finitely satisfiable. | {{Explain|Missing explicit reference to the use of Boolean Prime Ideal Theorem/Ultrafilter Lemma or Axiom of Choice/Zorn's Lemma}}
By definition, $T$ is finitely satisfiable means that every finite subset of $T$ is satisfiable.
Because the direction:
:$T$ satisfiable implies $T$ finitely satisfiable
is trivial, the pro... | Let $\LL$ be the [[Definition:Language of Predicate Logic|language of predicate logic]].
Let $T$ be a set of $\LL$-[[Definition:Sentence|sentences]].
Then $T$ is [[Definition:Satisfiable Set of Formulas|satisfiable]] {{iff}} $T$ is [[Definition:Finitely Satisfiable|finitely satisfiable]]. | {{Explain|Missing explicit reference to the use of Boolean Prime Ideal Theorem/Ultrafilter Lemma or Axiom of Choice/Zorn's Lemma}}
By definition, $T$ is [[Definition:Finitely Satisfiable|finitely satisfiable]] means that every [[Definition:Finite Subset|finite subset]] of $T$ is [[Definition:Satisfiable|satisfiable]].
... | Compactness Theorem | https://proofwiki.org/wiki/Compactness_Theorem | https://proofwiki.org/wiki/Compactness_Theorem | [
"Compactness Theorem",
"Model Theory for Predicate Logic",
"Mathematical Logic",
"Named Theorems"
] | [
"Definition:Language of Predicate Logic",
"Definition:Classes of WFFs/Sentence",
"Definition:Satisfiable/Set of Formulas",
"Definition:Finitely Satisfiable"
] | [
"Definition:Finitely Satisfiable",
"Definition:Finite Subset",
"Definition:Satisfiable",
"Definition:Satisfiable/Set of Formulas",
"Definition:Finitely Satisfiable",
"Definition:Converse",
"Definition:Finitely Satisfiable",
"Definition:Satisfiable/Set of Formulas"
] |
proofwiki-3750 | Compactness Theorem | Let $\LL$ be the language of predicate logic.
Let $T$ be a set of $\LL$-sentences.
Then $T$ is satisfiable {{iff}} $T$ is finitely satisfiable. | Suppose $\mathbf H$ does '''not''' have a model.
By the Main Lemma of Propositional Tableaux, $\mathbf H$ has a tableau confutation $T$.
By Tableau Confutation contains Finite Tableau Confutation, $T$ may be assumed to be finite.
Hence the set $\mathbf H'$ of all WFFs in $\mathbf H$ used somewhere in $T$ is finite.
Now... | Let $\LL$ be the [[Definition:Language of Predicate Logic|language of predicate logic]].
Let $T$ be a set of $\LL$-[[Definition:Sentence|sentences]].
Then $T$ is [[Definition:Satisfiable Set of Formulas|satisfiable]] {{iff}} $T$ is [[Definition:Finitely Satisfiable|finitely satisfiable]]. | Suppose $\mathbf H$ does '''not''' have a [[Definition:Model (Boolean Interpretations)|model]].
By the [[Main Lemma of Propositional Tableaux]], $\mathbf H$ has a [[Definition:Tableau Confutation|tableau confutation]] $T$.
By [[Tableau Confutation contains Finite Tableau Confutation]], $T$ may be assumed to be [[Defi... | Compactness Theorem for Boolean Interpretations/Proof 1 | https://proofwiki.org/wiki/Compactness_Theorem | https://proofwiki.org/wiki/Compactness_Theorem_for_Boolean_Interpretations/Proof_1 | [
"Compactness Theorem",
"Model Theory for Predicate Logic",
"Mathematical Logic",
"Named Theorems"
] | [
"Definition:Language of Predicate Logic",
"Definition:Classes of WFFs/Sentence",
"Definition:Satisfiable/Set of Formulas",
"Definition:Finitely Satisfiable"
] | [
"Definition:Model (Boolean Interpretations)",
"Main Lemma of Propositional Tableaux",
"Definition:Tableau Confutation",
"Tableau Confutation contains Finite Tableau Confutation",
"Definition:Propositional Tableau/Construction/Finite",
"Definition:Language of Propositional Logic/Formal Grammar/WFF",
"Def... |
proofwiki-3751 | Compactness Theorem | Let $\LL$ be the language of predicate logic.
Let $T$ be a set of $\LL$-sentences.
Then $T$ is satisfiable {{iff}} $T$ is finitely satisfiable. | If $\mathbf H$ is finite, the result is trivial.
So let $\mathbf H = \set {\mathbf A_n: n \in \N}$ be an enumeration of $\mathbf H$.
Define $\mathbf H_m = \set {\mathbf A_n: n \le m}$.
Let $T_1$ be the propositional tableau consisting of only a root node with hypothesis set $\mathbf H_1$.
For each $m \in \N$, construct... | Let $\LL$ be the [[Definition:Language of Predicate Logic|language of predicate logic]].
Let $T$ be a set of $\LL$-[[Definition:Sentence|sentences]].
Then $T$ is [[Definition:Satisfiable Set of Formulas|satisfiable]] {{iff}} $T$ is [[Definition:Finitely Satisfiable|finitely satisfiable]]. | If $\mathbf H$ is [[Definition:Finite Set|finite]], the result is trivial.
So let $\mathbf H = \set {\mathbf A_n: n \in \N}$ be an [[Definition:Enumeration|enumeration]] of $\mathbf H$.
Define $\mathbf H_m = \set {\mathbf A_n: n \le m}$.
Let $T_1$ be the [[Definition:Propositional Tableau|propositional tableau]] co... | Compactness Theorem for Boolean Interpretations/Proof 2 | https://proofwiki.org/wiki/Compactness_Theorem | https://proofwiki.org/wiki/Compactness_Theorem_for_Boolean_Interpretations/Proof_2 | [
"Compactness Theorem",
"Model Theory for Predicate Logic",
"Mathematical Logic",
"Named Theorems"
] | [
"Definition:Language of Predicate Logic",
"Definition:Classes of WFFs/Sentence",
"Definition:Satisfiable/Set of Formulas",
"Definition:Finitely Satisfiable"
] | [
"Definition:Finite Set",
"Definition:Enumeration",
"Definition:Propositional Tableau",
"Definition:Rooted Tree/Root Node",
"Definition:Labeled Tree for Propositional Logic/Hypothesis Set",
"Tableau Extension Lemma",
"Definition:Sequence",
"Definition:Finished Propositional Tableau",
"Definition:Exha... |
proofwiki-3752 | Compactness Theorem | Let $\LL$ be the language of predicate logic.
Let $T$ be a set of $\LL$-sentences.
Then $T$ is satisfiable {{iff}} $T$ is finitely satisfiable. | By Extend Theory to Satisfy Witness Property, there exist a language $\LL^*$ and a set of $\LL^*$-sentences $T^*$ satisfying:
* $T^*$ is finitely satisfiable
* If $T^*$ is satisfiable, then $T$ is satisfiable
* For every $\LL^*$-WFF of $1$ free variable $\map \phi x$, there exists some constant $c_\phi$ such that:
::$T... | Let $\LL$ be the [[Definition:Language of Predicate Logic|language of predicate logic]].
Let $T$ be a set of $\LL$-[[Definition:Sentence|sentences]].
Then $T$ is [[Definition:Satisfiable Set of Formulas|satisfiable]] {{iff}} $T$ is [[Definition:Finitely Satisfiable|finitely satisfiable]]. | By [[Extend Theory to Satisfy Witness Property]], there exist a [[Definition:Language of Predicate Logic|language]] $\LL^*$ and a [[Definition:Set|set]] of $\LL^*$-[[Definition:Sentence|sentences]] $T^*$ satisfying:
* $T^*$ is [[Definition:Finitely Satisfiable|finitely satisfiable]]
* If $T^*$ is [[Definition:Satisfiab... | Compactness Theorem/Proof using Consistency Principle | https://proofwiki.org/wiki/Compactness_Theorem | https://proofwiki.org/wiki/Compactness_Theorem/Proof_using_Consistency_Principle | [
"Compactness Theorem",
"Model Theory for Predicate Logic",
"Mathematical Logic",
"Named Theorems"
] | [
"Definition:Language of Predicate Logic",
"Definition:Classes of WFFs/Sentence",
"Definition:Satisfiable/Set of Formulas",
"Definition:Finitely Satisfiable"
] | [
"Extend Theory to Satisfy Witness Property",
"Definition:Language of Predicate Logic",
"Definition:Set",
"Definition:Classes of WFFs/Sentence",
"Definition:Finitely Satisfiable",
"Definition:Satisfiable",
"Definition:Satisfiable",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:... |
proofwiki-3753 | Compactness Theorem | Let $\LL$ be the language of predicate logic.
Let $T$ be a set of $\LL$-sentences.
Then $T$ is satisfiable {{iff}} $T$ is finitely satisfiable. | This proof is by contraposition.
The idea is to exploit the finiteness of proofs and the relation between satisfiability and deducibility to show that if $T$ is not satisfiable, then it must have a finite subset which can be used to prove to a contradiction.
Suppose $T$ is not satisfiable.
Since $T$ has no models, it v... | Let $\LL$ be the [[Definition:Language of Predicate Logic|language of predicate logic]].
Let $T$ be a set of $\LL$-[[Definition:Sentence|sentences]].
Then $T$ is [[Definition:Satisfiable Set of Formulas|satisfiable]] {{iff}} $T$ is [[Definition:Finitely Satisfiable|finitely satisfiable]]. | This proof is by [[Proof by Contraposition|contraposition]].
The idea is to exploit the finiteness of proofs and the relation between satisfiability and deducibility to show that if $T$ is not satisfiable, then it must have a finite subset which can be used to prove to a contradiction.
Suppose $T$ is not satisfiable.... | Compactness Theorem/Proof using Gödel's Completeness Theorem | https://proofwiki.org/wiki/Compactness_Theorem | https://proofwiki.org/wiki/Compactness_Theorem/Proof_using_Gödel's_Completeness_Theorem | [
"Compactness Theorem",
"Model Theory for Predicate Logic",
"Mathematical Logic",
"Named Theorems"
] | [
"Definition:Language of Predicate Logic",
"Definition:Classes of WFFs/Sentence",
"Definition:Satisfiable/Set of Formulas",
"Definition:Finitely Satisfiable"
] | [
"Proof by Contraposition",
"Gödel's Completeness Theorem",
"Soundness of First-Order Logic",
"Rule of Transposition"
] |
proofwiki-3754 | Compactness Theorem | Let $\LL$ be the language of predicate logic.
Let $T$ be a set of $\LL$-sentences.
Then $T$ is satisfiable {{iff}} $T$ is finitely satisfiable. | This proof actually demonstrates a stronger form of the Compactness Theorem by showing:
:If $T$ is finitely satisfiable and $\kappa$ is an infinite cardinal such that $\kappa > \size \LL$, then $T$ is satisfiable by a model whose cardinality is at most $\kappa$.
This is stronger than the original statement, since it pr... | Let $\LL$ be the [[Definition:Language of Predicate Logic|language of predicate logic]].
Let $T$ be a set of $\LL$-[[Definition:Sentence|sentences]].
Then $T$ is [[Definition:Satisfiable Set of Formulas|satisfiable]] {{iff}} $T$ is [[Definition:Finitely Satisfiable|finitely satisfiable]]. | This proof actually demonstrates a stronger form of the [[Compactness Theorem]] by showing:
:If $T$ is finitely satisfiable and $\kappa$ is an infinite cardinal such that $\kappa > \size \LL$, then $T$ is satisfiable by a model whose cardinality is at most $\kappa$.
This is stronger than the original statement, since ... | Compactness Theorem/Proof using Henkin Construction | https://proofwiki.org/wiki/Compactness_Theorem | https://proofwiki.org/wiki/Compactness_Theorem/Proof_using_Henkin_Construction | [
"Compactness Theorem",
"Model Theory for Predicate Logic",
"Mathematical Logic",
"Named Theorems"
] | [
"Definition:Language of Predicate Logic",
"Definition:Classes of WFFs/Sentence",
"Definition:Satisfiable/Set of Formulas",
"Definition:Finitely Satisfiable"
] | [
"Compactness Theorem",
"Extend Theory to Satisfy Witness Property",
"Definition:Witness Property",
"Finitely Satisfiable Theory has Maximal Finitely Satisfiable Extension",
"Definition:Witness Property",
"Maximal Finitely Satisfiable Theory with Witness Property is Satisfiable"
] |
proofwiki-3755 | Compactness Theorem | Let $\LL$ be the language of predicate logic.
Let $T$ be a set of $\LL$-sentences.
Then $T$ is satisfiable {{iff}} $T$ is finitely satisfiable. | The idea is to construct an ultraproduct using a purposefully selected ultrafilter and collection of models so that each sentence in $T$ will be realized as a result of Łoś's Theorem.
Let $\Sigma$ be the set of all finite subsets of $T$.
For every $\Sigma_0 \in \Sigma$, define:
:$F_{\Sigma_0} = \set {\Delta \in \Sigma:... | Let $\LL$ be the [[Definition:Language of Predicate Logic|language of predicate logic]].
Let $T$ be a set of $\LL$-[[Definition:Sentence|sentences]].
Then $T$ is [[Definition:Satisfiable Set of Formulas|satisfiable]] {{iff}} $T$ is [[Definition:Finitely Satisfiable|finitely satisfiable]]. | The idea is to construct an [[Definition:Ultraproduct|ultraproduct]] using a purposefully selected [[Definition:Ultrafilter|ultrafilter]] and collection of models so that each sentence in $T$ will be realized as a result of [[Łoś's Theorem]].
Let $\Sigma$ be the [[Definition:Set|set]] of all [[Definition:Finite Subset... | Compactness Theorem/Proof using Ultraproducts | https://proofwiki.org/wiki/Compactness_Theorem | https://proofwiki.org/wiki/Compactness_Theorem/Proof_using_Ultraproducts | [
"Compactness Theorem",
"Model Theory for Predicate Logic",
"Mathematical Logic",
"Named Theorems"
] | [
"Definition:Language of Predicate Logic",
"Definition:Classes of WFFs/Sentence",
"Definition:Satisfiable/Set of Formulas",
"Definition:Finitely Satisfiable"
] | [
"Definition:Ultraproduct",
"Definition:Ultrafilter on Set",
"Łoś's Theorem",
"Definition:Set",
"Definition:Finite Subset",
"Definition:Finite Subset",
"Definition:Subset",
"Definition:Finite Intersection Property",
"Łoś's Theorem"
] |
proofwiki-3756 | Łoś's Theorem | Let $\LL$ be a language.
Let $I$ be an infinite set.
Let $\UU$ be an ultrafilter on $I$.
Let $\map \phi {v_1, \ldots, v_n}$ be an $\LL$-formula.
Let $\MM$ be the ultraproduct:
:$\ds \paren {\prod_{i \mathop \in I} \MM_i} / \UU$
where each $\MM_i$ is an $\LL$-structure.
Then, for all $m_1 = \paren {m_{1, i} }_\UU, \dots... | We prove the $\LL$-sentences case by induction on the complexity of formulas. The general case trivially follows this proof.
We appeal to the interpretations of language symbols in the ultraproduct when viewed as an $\LL$-structure, the properties of ultrafilters, and make use of the Axiom of Choice.
The theorem holds... | Let $\LL$ be a [[Definition:Language of Predicate Logic|language]].
Let $I$ be an infinite set.
Let $\UU$ be an [[Definition:Ultrafilter on Set|ultrafilter]] on $I$.
Let $\map \phi {v_1, \ldots, v_n}$ be an $\LL$-[[Definition:Language of Predicate Logic/Formal Grammar|formula]].
Let $\MM$ be the [[Definition:Ultrap... | We prove the $\LL$-[[Definition:Classes of WFFs/Sentence|sentences]] case by induction on the complexity of formulas. The general case trivially follows this proof.
We appeal to the interpretations of language symbols in the ultraproduct when viewed as an $\LL$-structure, the properties of ultrafilters, and make use ... | Łoś's Theorem | https://proofwiki.org/wiki/Łoś's_Theorem | https://proofwiki.org/wiki/Łoś's_Theorem | [
"Model Theory for Predicate Logic"
] | [
"Definition:Language of Predicate Logic",
"Definition:Ultrafilter on Set",
"Definition:Language of Predicate Logic/Formal Grammar",
"Definition:Ultraproduct",
"Definition:Formal Semantics/Structure",
"Definition:Classes of WFFs/Sentence"
] | [
"Definition:Classes of WFFs/Sentence",
"Axiom:Axiom of Choice",
"Definition:Biconditional",
"Axiom:Axiom of Choice",
"Definition:Biconditional",
"Category:Model Theory for Predicate Logic"
] |
proofwiki-3757 | Closed Extension Topology is Topology | Let $T = \struct {S, \tau}$ be a topological space.
Let $\tau^*_p$ be the closed extension topology of $\tau$.
Then $\tau^*_p$ is a topology on $S^*_p = S \cup \set p$. | By definition:
:$\tau^*_p = \set {U \cup \set p: U \in \tau} \cup \set \O$
We have that $\O \in \tau^*_p$ by definition.
We also have that $S \in \tau$ so $S \cup \set p \in \tau^*_p$.
Now let $U_1, U_2 \in \tau^*_p$.
Then $U_1^* = U_1 \setminus \set p \in \tau, U_2^* = U_2 \setminus \set p \in \tau$.
Then $p \in U_1$ ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\tau^*_p$ be the [[Definition:Closed Extension Topology|closed extension topology]] of $\tau$.
Then $\tau^*_p$ is a [[Definition:Topology|topology]] on $S^*_p = S \cup \set p$. | By definition:
:$\tau^*_p = \set {U \cup \set p: U \in \tau} \cup \set \O$
We have that $\O \in \tau^*_p$ by definition.
We also have that $S \in \tau$ so $S \cup \set p \in \tau^*_p$.
Now let $U_1, U_2 \in \tau^*_p$.
Then $U_1^* = U_1 \setminus \set p \in \tau, U_2^* = U_2 \setminus \set p \in \tau$.
Then $p \i... | Closed Extension Topology is Topology | https://proofwiki.org/wiki/Closed_Extension_Topology_is_Topology | https://proofwiki.org/wiki/Closed_Extension_Topology_is_Topology | [
"Closed Extension Topologies"
] | [
"Definition:Topological Space",
"Definition:Closed Extension Topology",
"Definition:Topology"
] | [
"Definition:Topology"
] |
proofwiki-3758 | Tarski-Vaught Test | Let $\MM, \NN$ be $\LL$-structures such that $\MM$ is a substructure of $\NN$.
{{wtd|The page Definition:Structure is a disambiguation page, in which the form in which it is used here may not be included. The level of clarity in this page generally needs improving. Hence the invocation of the Disambiguate template.}}
{... | === Sufficient Condition ===
Let $\MM$ be an elementary substructure of $\NN$.
Then $\NN \models \exists x: \map \phi {x, \bar a}$ implies that $\MM \models \exists x: \map \phi {x, \bar a}$.
Hence there exists some $m$ in $\MM$ such that:
:$\MM \models \map \phi {m, \bar a}$.
Passing back up to $\NN$ yields the result... | Let $\MM, \NN$ be [[Definition:Structure|$\LL$-structures]] such that $\MM$ is a [[Definition:Substructure|substructure]] of $\NN$.
{{wtd|The page [[Definition:Structure]] is a disambiguation page, in which the form in which it is used here may not be included. The level of clarity in this page generally needs improvi... | === Sufficient Condition ===
Let $\MM$ be an elementary substructure of $\NN$.
Then $\NN \models \exists x: \map \phi {x, \bar a}$ implies that $\MM \models \exists x: \map \phi {x, \bar a}$.
Hence there exists some $m$ in $\MM$ such that:
:$\MM \models \map \phi {m, \bar a}$.
Passing back up to $\NN$ yields the re... | Tarski-Vaught Test | https://proofwiki.org/wiki/Tarski-Vaught_Test | https://proofwiki.org/wiki/Tarski-Vaught_Test | [
"Model Theory for Predicate Logic"
] | [
"Definition:Structure",
"Definition:Substructure",
"Definition:Structure",
"Definition:Elementary Substructure",
"Definition:Logical Formula",
"Definition:Substructure",
"Definition:Elementary Substructure",
"Definition:Structure",
"Definition:Elementary Substructure",
"Definition:Logical Formula"... | [] |
proofwiki-3759 | Quantifier Free Formula is Preserved by Superstructure | Let $\MM, \NN$ be $\LL$-structures such that $\MM$ is a substructure of $\NN$.
Let $\map \phi {\bar x}$ be a quantifier-free $\LL$-formula, and let $\bar a \in\MM$.
Then $\MM \models \map \phi {\bar a}$ {{iff}} $\NN \models \map \phi {\bar a}$. | The proof is done by induction on complexity of formulas.
Note that since interpretations of terms with parameters from $\MM$ are preserved when passing to superstructures, we have that $\map {t^\MM} {\bar a} = \map {t^\NN} {\bar a}$ whenever $t$ is an $\LL$-term and $\bar a$ is in $\MM$.
First, we verify the theorem f... | Let $\MM, \NN$ be $\LL$-[[Definition:First-Order Structure|structures]] such that $\MM$ is a [[Definition:Substructure|substructure]] of $\NN$.
Let $\map \phi {\bar x}$ be a [[Definition:Quantifier-Free Formula|quantifier-free $\LL$-formula]], and let $\bar a \in\MM$.
Then $\MM \models \map \phi {\bar a}$ {{iff}} $... | The proof is done by induction on complexity of formulas.
Note that since [[Interpretations of Terms are Preserved by Superstructures|interpretations of terms with parameters from $\MM$ are preserved when passing to superstructures]], we have that $\map {t^\MM} {\bar a} = \map {t^\NN} {\bar a}$ whenever $t$ is an $\L... | Quantifier Free Formula is Preserved by Superstructure | https://proofwiki.org/wiki/Quantifier_Free_Formula_is_Preserved_by_Superstructure | https://proofwiki.org/wiki/Quantifier_Free_Formula_is_Preserved_by_Superstructure | [
"Model Theory for Predicate Logic"
] | [
"Definition:Structure for Predicate Logic",
"Definition:Substructure",
"Definition:Quantifier-Free Formula"
] | [
"Interpretations of Terms are Preserved by Superstructures",
"Definition:Term",
"Category:Model Theory for Predicate Logic"
] |
proofwiki-3760 | Algebraically Closed Field is Infinite | Let $F$ be an algebraically closed field.
Then $F$ is infinite. | We prove the contrapositive: that a Galois field cannot be algebraically closed.
Let $F$ be Galois.
Define the polynomial:
:$\ds \map f x = 1 + \prod_{a \mathop \in F} \paren {x - a}$
By definition, a field is a ring.
Thus by Ring Product with Zero:
:$\ds \forall x \in F: \prod_{a \mathop \in F} \paren {x - a} = 0$
But... | Let $F$ be an [[Definition:Algebraically Closed Field|algebraically closed field]].
Then $F$ is [[Definition:Infinite Field|infinite]]. | We prove the [[Definition:Contrapositive Statement|contrapositive]]: that a [[Definition:Galois Field|Galois field]] cannot be [[Definition:Algebraically Closed Field|algebraically closed]].
Let $F$ be [[Definition:Galois Field|Galois]].
Define the [[Definition:Polynomial (Abstract Algebra)|polynomial]]:
:$\ds \map f... | Algebraically Closed Field is Infinite | https://proofwiki.org/wiki/Algebraically_Closed_Field_is_Infinite | https://proofwiki.org/wiki/Algebraically_Closed_Field_is_Infinite | [
"Field Theory"
] | [
"Definition:Algebraically Closed Field",
"Definition:Infinite Field"
] | [
"Definition:Contrapositive Statement",
"Definition:Galois Field",
"Definition:Algebraically Closed Field",
"Definition:Galois Field",
"Definition:Polynomial over Ring",
"Definition:Field (Abstract Algebra)",
"Definition:Ring (Abstract Algebra)",
"Ring Product with Zero",
"Definition:Root of Polynomi... |
proofwiki-3761 | Field of Uncountable Cardinality K has Transcendence Degree K | Let $F$ be a field of uncountable cardinality $\kappa$.
Then $F$ has transcendence degree $\kappa$ over its prime field. | We prove the theorem for fields with characteristic $p = 0$.
In the case where $p$ is a prime, the proof is similar, but instead we view the fields as extensions of $\Z / \Z_p$.
The main idea is to exploit the lower cardinality of sets of polynomials in order to keep finding algebraically independent elements of $F$.
S... | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]] of [[Definition:Uncountable Set|uncountable]] [[Definition:Cardinality|cardinality]] $\kappa$.
Then $F$ has [[Definition:Transcendence Degree|transcendence degree]] $\kappa$ over its [[Definition:Prime Subfield|prime field]]. | We prove the theorem for fields with characteristic $p = 0$.
In the case where $p$ is a prime, the proof is similar, but instead we view the fields as extensions of $\Z / \Z_p$.
The main idea is to exploit the lower cardinality of sets of polynomials in order to keep finding algebraically independent elements of $F$.... | Field of Uncountable Cardinality K has Transcendence Degree K | https://proofwiki.org/wiki/Field_of_Uncountable_Cardinality_K_has_Transcendence_Degree_K | https://proofwiki.org/wiki/Field_of_Uncountable_Cardinality_K_has_Transcendence_Degree_K | [
"Field Theory",
"Uncountable Sets"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Uncountable/Set",
"Definition:Cardinality",
"Definition:Transcendence Degree",
"Definition:Prime Subfield"
] | [
"Definition:Characteristic of Field",
"Definition:Field Extension",
"Definition:Algebraically Independent",
"Definition:Transfinite Induction",
"Axiom:Axiom of Choice",
"Definition:Ordinal",
"Definition:Set",
"Definition:Singleton",
"Algebraic Numbers are Countable",
"Definition:Successor Ordinal"... |
proofwiki-3762 | Theory of Algebraically Closed Fields of Characteristic p is Complete | Let $p$ be either $0$ or a prime number.
Let $ACF_p$ be the theory of algebraically closed fields of characteristic $p$ in the language $\LL_r = \set {0, 1, +, -, \cdot}$ for rings, where:
:$0, 1$ are constants
and:
:$+, -, \cdot$ are binary functions.
Then $ACF_p$ is complete. | By the Łoś-Vaught Test, it suffices to show that $ACF_p$ is satisfiable, has no finite models, and is $\kappa$-categorical for some uncountable $\kappa$. | Let $p$ be either $0$ or a [[Definition:Prime Number|prime number]].
Let $ACF_p$ be the [[Definition:Theory of Elementary Class|theory]] of [[Definition:Algebraically Closed Field|algebraically closed fields]] of [[Definition:Characteristic of Field|characteristic]] $p$ in the language $\LL_r = \set {0, 1, +, -, \cdot... | By the [[Łoś-Vaught Test]], it suffices to show that $ACF_p$ is satisfiable, has no finite models, and is [[Definition:Categorical (Model Theory)|$\kappa$-categorical]] for some [[Definition:Uncountable Set|uncountable]] $\kappa$. | Theory of Algebraically Closed Fields of Characteristic p is Complete | https://proofwiki.org/wiki/Theory_of_Algebraically_Closed_Fields_of_Characteristic_p_is_Complete | https://proofwiki.org/wiki/Theory_of_Algebraically_Closed_Fields_of_Characteristic_p_is_Complete | [
"Model Theory for Predicate Logic"
] | [
"Definition:Prime Number",
"Definition:Theory of Elementary Class",
"Definition:Algebraically Closed Field",
"Definition:Characteristic of Field",
"Definition:Ring (Abstract Algebra)",
"Definition:Constant",
"Definition:Operation/Binary Operation",
"Definition:Complete Theory"
] | [
"Łoś-Vaught Test",
"Definition:Categorical (Model Theory)",
"Definition:Uncountable/Set"
] |
proofwiki-3763 | Lefschetz Principle (First-Order) | Let $\phi$ be a sentence in the language $\LL_r = \set {0, 1, +, -, \cdot}$ for rings, where $0, 1$ are constants and $+, -, \cdot$ are binary functions.
{{TFAE}}
:$(1): \quad \phi$ is true in every algebraically closed field of characteristic $0$.
:$(2): \quad \phi$ is true in some algebraically closed field of charac... | === $(1)$ iff $(2)$ ===
From Theory of Algebraically Closed Fields of Characteristic $p$ is Complete:
the theory $ACF_p$ of algebraically closed fields of characteristic $p$ is complete.
That is, all such fields satisfy the exact same $\LL_r$ sentences.
{{qed|lemma}} | Let $\phi$ be a sentence in the language $\LL_r = \set {0, 1, +, -, \cdot}$ for rings, where $0, 1$ are constants and $+, -, \cdot$ are binary functions.
{{TFAE}}
:$(1): \quad \phi$ is true in every algebraically closed field of characteristic $0$.
:$(2): \quad \phi$ is true in some algebraically closed field of cha... | === $(1)$ iff $(2)$ ===
From [[Theory of Algebraically Closed Fields of Characteristic p is Complete|Theory of Algebraically Closed Fields of Characteristic $p$ is Complete]]:
the theory $ACF_p$ of algebraically closed fields of characteristic $p$ is complete.
That is, all such fields satisfy the exact same $\LL_r$ ... | Lefschetz Principle (First-Order) | https://proofwiki.org/wiki/Lefschetz_Principle_(First-Order) | https://proofwiki.org/wiki/Lefschetz_Principle_(First-Order) | [
"Algebraic Geometry",
"Field Theory",
"Model Theory for Predicate Logic",
"Proofs by Contraposition"
] | [] | [
"Theory of Algebraically Closed Fields of Characteristic p is Complete"
] |
proofwiki-3764 | CNF Satisfiability Problem is NP-Complete | The conjunctive normal form boolean satisfiability problem (CNF SAT) is NP-complete. | Let $P$ be a CNF SAT problem. | The [[Definition:CNF Satisfiability Problem|conjunctive normal form boolean satisfiability problem]] (CNF SAT) is [[Definition:NP-Complete|NP-complete]]. | Let $P$ be a [[Definition:CNF Satisfiability Problem|CNF SAT problem]]. | CNF Satisfiability Problem is NP-Complete | https://proofwiki.org/wiki/CNF_Satisfiability_Problem_is_NP-Complete | https://proofwiki.org/wiki/CNF_Satisfiability_Problem_is_NP-Complete | [
"Mathematical Logic"
] | [
"Definition:CNF Satisfiability Problem",
"Definition:NP-Complete"
] | [
"Definition:CNF Satisfiability Problem",
"Definition:CNF Satisfiability Problem",
"Definition:CNF Satisfiability Problem",
"Definition:CNF Satisfiability Problem",
"Definition:CNF Satisfiability Problem",
"Definition:CNF Satisfiability Problem"
] |
proofwiki-3765 | Closed Sets of Closed Extension Topology | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.
Then the closed sets of $T^*_p$ (apart from $S^*_p$) are the closed sets of $T$.
This explains why $\tau^*_p$ is called the closed extension topology of $\tau$. | By definition:
:$\tau^*_p = \set {U \cup \set p: U \in \tau} \cup \set \O$
Let $V \subseteq S^*_p$ be closed in $T^*_p$.
Then $S^*_p \setminus V$ is open in $T^*_p$.
Then $\struct {S^*_p \setminus V} \setminus \set p$ is open in $T$.
From Set Difference with Union we have:
:$\struct {S^*_p \setminus V} \setminus \set p... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the [[Definition:Closed Extension Space|closed extension space]] of $T$.
Then the [[Definition:Closed Set (Topology)|closed sets]] of $T^*_p$ (apart from $S^*_p$) are the [[Definition:Close... | By definition:
:$\tau^*_p = \set {U \cup \set p: U \in \tau} \cup \set \O$
Let $V \subseteq S^*_p$ be [[Definition:Closed Set (Topology)|closed]] in $T^*_p$.
Then $S^*_p \setminus V$ is [[Definition:Open Set (Topology)|open]] in $T^*_p$.
Then $\struct {S^*_p \setminus V} \setminus \set p$ is [[Definition:Open Set ... | Closed Sets of Closed Extension Topology | https://proofwiki.org/wiki/Closed_Sets_of_Closed_Extension_Topology | https://proofwiki.org/wiki/Closed_Sets_of_Closed_Extension_Topology | [
"Closed Extension Topologies"
] | [
"Definition:Topological Space",
"Definition:Closed Extension Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Extension Topology"
] | [
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Set Difference with Union"
] |
proofwiki-3766 | Particular Point Topology is Closed Extension Topology of Discrete Topology | Let $S$ be a set and let $p \in S$.
Let $\tau_p$ be the particular point topology on $S$.
Let $T = \struct {S \setminus \set p, \vartheta}$ be the discrete topological space on $S \setminus \set p$.
Then $T^* = \struct {S, \tau_p}$ is a closed extension space of $T$. | Directly apparent from the definitions of particular point topology, discrete topological space and closed extension space.
{{qed}} | Let $S$ be a [[Definition:Set|set]] and let $p \in S$.
Let $\tau_p$ be the [[Definition:Particular Point Topology|particular point topology]] on $S$.
Let $T = \struct {S \setminus \set p, \vartheta}$ be the [[Definition:Discrete Space|discrete topological space]] on $S \setminus \set p$.
Then $T^* = \struct {S, \t... | Directly apparent from the definitions of [[Definition:Particular Point Topology|particular point topology]], [[Definition:Discrete Space|discrete topological space]] and [[Definition:Closed Extension Space|closed extension space]].
{{qed}} | Particular Point Topology is Closed Extension Topology of Discrete Topology | https://proofwiki.org/wiki/Particular_Point_Topology_is_Closed_Extension_Topology_of_Discrete_Topology | https://proofwiki.org/wiki/Particular_Point_Topology_is_Closed_Extension_Topology_of_Discrete_Topology | [
"Closed Extension Topologies",
"Particular Point Topologies",
"Discrete Topologies"
] | [
"Definition:Set",
"Definition:Particular Point Topology",
"Definition:Discrete Topology",
"Definition:Closed Extension Topology"
] | [
"Definition:Particular Point Topology",
"Definition:Discrete Topology",
"Definition:Closed Extension Topology"
] |
proofwiki-3767 | Closed Extension Topology is not T1 | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.
Then $T^*_p$ is not a $T_1$ space. | By definition:
:$\tau^*_p = \set {U \cup \set p: U \in \tau} \cup \set \O$
Let $x \in S^*_p, x \ne p$.
Let $U = \set p$.
Then $U \in \tau^*_p$ such that $p \in U, x \notin U$.
But there is no $V \in \tau^*_p$ such that $x \in V, p \notin V$, by definition of the closed extension topology.
Hence $T^*_p$ can not be a $T_... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the [[Definition:Closed Extension Space|closed extension space]] of $T$.
Then $T^*_p$ is not a [[Definition:T1 Space|$T_1$ space]]. | By definition:
:$\tau^*_p = \set {U \cup \set p: U \in \tau} \cup \set \O$
Let $x \in S^*_p, x \ne p$.
Let $U = \set p$.
Then $U \in \tau^*_p$ such that $p \in U, x \notin U$.
But there is no $V \in \tau^*_p$ such that $x \in V, p \notin V$, by definition of the [[Definition:Closed Extension Topology|closed exten... | Closed Extension Topology is not T1 | https://proofwiki.org/wiki/Closed_Extension_Topology_is_not_T1 | https://proofwiki.org/wiki/Closed_Extension_Topology_is_not_T1 | [
"Closed Extension Topologies",
"Examples of T1 Spaces"
] | [
"Definition:Topological Space",
"Definition:Closed Extension Topology",
"Definition:T1 Space"
] | [
"Definition:Closed Extension Topology",
"Definition:T1 Space"
] |
proofwiki-3768 | Algebraic Numbers are Countable | The set $\Bbb A$ of algebraic numbers is countable. | By definition, $\Bbb A$ is the subset of the complex numbers which consists of roots of polynomials with coefficients in $\Q$.
We can prove the theorem by a cardinality argument, counting the number of such polynomials and roots.
By Set of Polynomials over Infinite Set has Same Cardinality, the set $\Q \sqbrk x$ of pol... | The [[Definition:Set|set]] $\Bbb A$ of [[Definition:Algebraic Number|algebraic numbers]] is [[Definition:Countable Set|countable]]. | By definition, $\Bbb A$ is the [[Definition:Subset|subset]] of the [[Definition:Complex Number|complex numbers]] which consists of [[Definition:Root of Polynomial|roots of polynomials]] with coefficients in $\Q$.
We can prove the theorem by a cardinality argument, counting the number of such polynomials and roots.
By... | Algebraic Numbers are Countable | https://proofwiki.org/wiki/Algebraic_Numbers_are_Countable | https://proofwiki.org/wiki/Algebraic_Numbers_are_Countable | [
"Countable Sets",
"Polynomial Theory",
"Algebraic Numbers"
] | [
"Definition:Set",
"Definition:Algebraic Number",
"Definition:Countable Set"
] | [
"Definition:Subset",
"Definition:Complex Number",
"Definition:Root of Polynomial",
"Set of Polynomials over Infinite Set has Same Cardinality",
"Definition:Set",
"Definition:Countable Set",
"Definition:Set Union",
"Polynomial over Field has Finitely Many Roots",
"Definition:Set Union/Countable Union... |
proofwiki-3769 | Limit Points in Closed Extension Space | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.
Let $x \in S$.
Then $x$ is a limit point of $p$. | Every open set of $T^*_p = \struct {S^*_p, \tau^*_p}$ except $\O$ contains the point $p$ by definition.
So every open set $U \in \tau^*_p$ such that $x \in U$ contains $p$.
So by definition of the limit point of a point, $x$ is a limit point of $p$.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the [[Definition:Closed Extension Space|closed extension space]] of $T$.
Let $x \in S$.
Then $x$ is a [[Definition:Limit Point of Point|limit point]] of $p$. | Every [[Definition:Open Set (Topology)|open set]] of $T^*_p = \struct {S^*_p, \tau^*_p}$ except $\O$ contains the point $p$ by [[Definition:Closed Extension Topology|definition]].
So every [[Definition:Open Set (Topology)|open set]] $U \in \tau^*_p$ such that $x \in U$ contains $p$.
So by definition of the [[Definiti... | Limit Points in Closed Extension Space | https://proofwiki.org/wiki/Limit_Points_in_Closed_Extension_Space | https://proofwiki.org/wiki/Limit_Points_in_Closed_Extension_Space | [
"Limit Points in Closed Extension Space",
"Closed Extension Topologies",
"Examples of Limit Points"
] | [
"Definition:Topological Space",
"Definition:Closed Extension Topology",
"Definition:Limit Point/Topology/Point"
] | [
"Definition:Open Set/Topology",
"Definition:Closed Extension Topology",
"Definition:Open Set/Topology",
"Definition:Limit Point/Topology/Point",
"Definition:Limit Point/Topology/Point"
] |
proofwiki-3770 | Closure of Open Set of Closed Extension Space | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.
Then:
:$U^- = S^*_p$
where $U^-$ denotes the closure of $U$ in $T^*_p$. | By definition, $\forall U \in \tau^*_p, u \ne \O: p \in U$.
From Limit Points in Closed Extension Space, every point in $S^*_p$ is a limit point of $p$.
So by definition of closure, every point in $S^*_p$ is in $U^-$.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the [[Definition:Closed Extension Space|closed extension space]] of $T$.
Then:
:$U^- = S^*_p$
where $U^-$ denotes the [[Definition:Closure (Topology)|closure]] of $U$ in $T^*_p$. | By definition, $\forall U \in \tau^*_p, u \ne \O: p \in U$.
From [[Limit Points in Closed Extension Space]], every point in $S^*_p$ is a [[Definition:Limit Point of Point|limit point of $p$]].
So by definition of [[Definition:Closure (Topology)|closure]], every point in $S^*_p$ is in $U^-$.
{{qed}} | Closure of Open Set of Closed Extension Space | https://proofwiki.org/wiki/Closure_of_Open_Set_of_Closed_Extension_Space | https://proofwiki.org/wiki/Closure_of_Open_Set_of_Closed_Extension_Space | [
"Closed Extension Topologies",
"Open Sets",
"Set Closures"
] | [
"Definition:Topological Space",
"Definition:Closed Extension Topology",
"Definition:Closure (Topology)"
] | [
"Limit Points in Closed Extension Space",
"Definition:Limit Point/Topology/Point",
"Definition:Closure (Topology)"
] |
proofwiki-3771 | Closed Extension Space is Irreducible | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.
Then $T^*_p$ is irreducible. | Trivially, by definition, every open set in $T^*_p$ contains $p$.
So:
:$\forall U_1, U_2 \in \tau^*_p: p \in U_1 \cap U_2$
for $U_1, U_2 \ne \O$.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the [[Definition:Closed Extension Space|closed extension space]] of $T$.
Then $T^*_p$ is [[Definition:Irreducible Space|irreducible]]. | Trivially, by definition, every [[Definition:Open Set (Topology)|open set]] in $T^*_p$ contains $p$.
So:
:$\forall U_1, U_2 \in \tau^*_p: p \in U_1 \cap U_2$
for $U_1, U_2 \ne \O$.
{{qed}} | Closed Extension Space is Irreducible/Proof 1 | https://proofwiki.org/wiki/Closed_Extension_Space_is_Irreducible | https://proofwiki.org/wiki/Closed_Extension_Space_is_Irreducible/Proof_1 | [
"Closed Extension Space is Irreducible",
"Closed Extension Topologies",
"Examples of Irreducible Spaces"
] | [
"Definition:Topological Space",
"Definition:Closed Extension Topology",
"Definition:Irreducible Space"
] | [
"Definition:Open Set/Topology"
] |
proofwiki-3772 | Closed Extension Space is Irreducible | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.
Then $T^*_p$ is irreducible. | From Closure of Open Set of Closed Extension Space we have that:
:$\forall U \in \tau^*_p: U \ne \O \implies U^- = S$
where $U^-$ is the closure of $U$.
The result then follows by definition of irreducible space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the [[Definition:Closed Extension Space|closed extension space]] of $T$.
Then $T^*_p$ is [[Definition:Irreducible Space|irreducible]]. | From [[Closure of Open Set of Closed Extension Space]] we have that:
:$\forall U \in \tau^*_p: U \ne \O \implies U^- = S$
where $U^-$ is the [[Definition:Closure (Topology)|closure]] of $U$.
The result then follows by definition of [[Definition:Irreducible Space/Definition 6|irreducible space]].
{{qed}} | Closed Extension Space is Irreducible/Proof 2 | https://proofwiki.org/wiki/Closed_Extension_Space_is_Irreducible | https://proofwiki.org/wiki/Closed_Extension_Space_is_Irreducible/Proof_2 | [
"Closed Extension Space is Irreducible",
"Closed Extension Topologies",
"Examples of Irreducible Spaces"
] | [
"Definition:Topological Space",
"Definition:Closed Extension Topology",
"Definition:Irreducible Space"
] | [
"Closure of Open Set of Closed Extension Space",
"Definition:Closure (Topology)",
"Definition:Irreducible Space/Definition 6"
] |
proofwiki-3773 | Polynomial over Field has Finitely Many Roots | Let $F$ be a field.
Let $F \left[{x}\right]$ be the ring of polynomial functions in the indeterminate $x$.
If $p \in F \left[{x}\right]$ be non-null, then $p$ has finitely many roots in $F$. | Let $n \ge 1$ be the degree of $p$.
We argue that $p$ has at most $n$ roots in $F$.
Let $A$ be the set of roots of $p$.
Let $a \in A$.
By the Polynomial Factor Theorem:
:$p \left({x}\right) = q_1 \left({x}\right) \cdot \left({x - a}\right)$
:where $\deg q_1 = n - 1$.
Let $a' \in A$ such that $a' \ne a$.
Then since:
:$p... | Let $F$ be a [[Definition:Field (Abstract Algebra)|field]].
Let $F \left[{x}\right]$ be the [[Definition:Ring of Polynomial Functions|ring of polynomial functions]] in the indeterminate $x$.
If $p \in F \left[{x}\right]$ be non-[[Definition:Null Polynomial over Ring|null]], then $p$ has finitely many roots in $F$. | Let $n \ge 1$ be the [[Definition:Degree (Polynomial)|degree]] of $p$.
We argue that $p$ has at most $n$ [[Definition:Root of Polynomial|roots]] in $F$.
Let $A$ be the [[Definition:Set|set]] of roots of $p$.
Let $a \in A$.
By the [[Polynomial Factor Theorem]]:
:$p \left({x}\right) = q_1 \left({x}\right) \cdot \lef... | Polynomial over Field has Finitely Many Roots | https://proofwiki.org/wiki/Polynomial_over_Field_has_Finitely_Many_Roots | https://proofwiki.org/wiki/Polynomial_over_Field_has_Finitely_Many_Roots | [
"Polynomial Theory"
] | [
"Definition:Field (Abstract Algebra)",
"Definition:Ring of Polynomial Functions",
"Definition:Null Polynomial/Ring"
] | [
"Definition:Degree of Polynomial",
"Definition:Root of Polynomial",
"Definition:Set",
"Polynomial Factor Theorem",
"Polynomial Factor Theorem",
"Category:Polynomial Theory"
] |
proofwiki-3774 | Set of Polynomials over Infinite Set has Same Cardinality | Let $S$ be a set of infinite cardinality $\kappa$.
Let $S \sqbrk x$ be the set of polynomial forms over $S$ in the indeterminate $x$.
Then $S \sqbrk x$ has cardinality $\kappa$. | Since $S \sqbrk x$ contains a copy of $S$ as constant polynomials, we have an injection $S \to S \sqbrk x$.
We define an injection from $S \sqbrk x$ to the set $\FF$ of finite sequences over $S$ as follows:
Each polynomial in $f \in S \sqbrk x$ is of the form:
:$f = a_0 + a_1 x + a_2 x^2 + \dotsb + a_n x^n$
where $a_n$... | Let $S$ be a [[Definition:Set|set]] of [[Definition:Infinite Set|infinite]] [[Definition:Cardinality|cardinality]] $\kappa$.
Let $S \sqbrk x$ be the set of [[Definition:Polynomial Form|polynomial forms]] over $S$ in the [[Definition:Indeterminate|indeterminate]] $x$.
Then $S \sqbrk x$ has [[Definition:Cardinality|ca... | Since $S \sqbrk x$ contains a copy of $S$ as [[Definition:Constant Polynomial|constant polynomials]], we have an [[Definition:Injection|injection]] $S \to S \sqbrk x$.
We define an [[Definition:Injection|injection]] from $S \sqbrk x$ to the [[Definition:Set|set]] $\FF$ of [[Definition:Finite Sequence|finite sequences... | Set of Polynomials over Infinite Set has Same Cardinality | https://proofwiki.org/wiki/Set_of_Polynomials_over_Infinite_Set_has_Same_Cardinality | https://proofwiki.org/wiki/Set_of_Polynomials_over_Infinite_Set_has_Same_Cardinality | [
"Polynomial Theory"
] | [
"Definition:Set",
"Definition:Infinite Set",
"Definition:Cardinality",
"Definition:Polynomial over Ring as Function on Free Monoid on Set",
"Definition:Indeterminate",
"Definition:Cardinality"
] | [
"Definition:Constant Polynomial",
"Definition:Injection",
"Definition:Injection",
"Definition:Set",
"Definition:Finite Sequence",
"Definition:Polynomial",
"Definition:Polynomial",
"Definition:Finite Sequence",
"Definition:Coefficient of Polynomial",
"Equality of Polynomials",
"Definition:Injecti... |
proofwiki-3775 | Excluded Point Topology is Topology | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $\tau_{\bar p}$ is a topology on $S$, and $T$ is a topological space. | We have by definition that $S \in \tau_{\bar p}$, and as $p \notin \O$ we have that $\O \in \tau_{\bar p}$.
Now let $U_1, U_2 \in \tau_{\bar p}$.
By definition $p \notin U_1$ and $p \notin U_2$.
By definition of set intersection:
:$p \notin U_1 \cap U_2$
So $U_1 \cap U_2 \in \tau_{\bar p}$.
Now let $\UU \subseteq \tau_... | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Space|excluded point space]].
Then $\tau_{\bar p}$ is a [[Definition:Topology|topology]] on $S$, and $T$ is a [[Definition:Topological Space|topological space]]. | We have [[Definition:Excluded Point Space|by definition]] that $S \in \tau_{\bar p}$, and as $p \notin \O$ we have that $\O \in \tau_{\bar p}$.
Now let $U_1, U_2 \in \tau_{\bar p}$.
By definition $p \notin U_1$ and $p \notin U_2$.
By definition of [[Definition:Set Intersection|set intersection]]:
:$p \notin U_1 \c... | Excluded Point Topology is Topology | https://proofwiki.org/wiki/Excluded_Point_Topology_is_Topology | https://proofwiki.org/wiki/Excluded_Point_Topology_is_Topology | [
"Excluded Point Topologies"
] | [
"Definition:Excluded Point Topology",
"Definition:Topology",
"Definition:Topological Space"
] | [
"Definition:Excluded Point Topology",
"Definition:Set Intersection",
"Set is Subset of Union",
"Definition:Topology"
] |
proofwiki-3776 | Open Extension Topology is Topology | Let $T = \struct {S, \tau}$ be a topological space.
Let $\tau^*_{\bar p}$ be the open extension topology of $\tau$.
Then $\tau^*_{\bar p}$ is a topology on $S^*_p = S \cup \set p$. | By definition:
:$\tau^*_{\bar p} = \set {U: U \in \tau} \cup \set {S^*_p}$
We have that $S^*_p \in \tau^*_{\bar p}$ by definition.
We also have that $\O \in \tau$ so $\O \in \tau^*_{\bar p}$.
Now let $U_1, U_2 \in \tau^*_{\bar p}$.
Then:
{{begin-eqn}}
{{eqn | l = U_1, U_2
| o = \in
| r = \tau
| c =
}... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $\tau^*_{\bar p}$ be the [[Definition:Open Extension Topology|open extension topology]] of $\tau$.
Then $\tau^*_{\bar p}$ is a [[Definition:Topology|topology]] on $S^*_p = S \cup \set p$. | By definition:
:$\tau^*_{\bar p} = \set {U: U \in \tau} \cup \set {S^*_p}$
We have that $S^*_p \in \tau^*_{\bar p}$ by definition.
We also have that $\O \in \tau$ so $\O \in \tau^*_{\bar p}$.
Now let $U_1, U_2 \in \tau^*_{\bar p}$.
Then:
{{begin-eqn}}
{{eqn | l = U_1, U_2
| o = \in
| r = \tau
|... | Open Extension Topology is Topology | https://proofwiki.org/wiki/Open_Extension_Topology_is_Topology | https://proofwiki.org/wiki/Open_Extension_Topology_is_Topology | [
"Open Extension Topologies"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:Topology"
] | [
"Definition:Topology"
] |
proofwiki-3777 | Excluded Point Topology is Open Extension Topology of Discrete Topology | Let $S$ be a set.
Let $p \in S$.
Let $\tau_{\bar p}$ be the excluded point topology on $S$.
Let $T = \struct {S \setminus \set p, \tau_D}$ be the discrete topological space on $S \setminus \set p$.
Then $T^* = \struct {S, \tau_{\bar p} }$ is an open extension space of $T$. | Directly apparent from the definitions of excluded point topology, discrete topological space and open extension space.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $p \in S$.
Let $\tau_{\bar p}$ be the [[Definition:Excluded Point Topology|excluded point topology]] on $S$.
Let $T = \struct {S \setminus \set p, \tau_D}$ be the [[Definition:Discrete Space|discrete topological space]] on $S \setminus \set p$.
Then $T^* = \struct {S, \tau_... | Directly apparent from the definitions of [[Definition:Excluded Point Topology|excluded point topology]], [[Definition:Discrete Space|discrete topological space]] and [[Definition:Open Extension Space|open extension space]].
{{qed}} | Excluded Point Topology is Open Extension Topology of Discrete Topology | https://proofwiki.org/wiki/Excluded_Point_Topology_is_Open_Extension_Topology_of_Discrete_Topology | https://proofwiki.org/wiki/Excluded_Point_Topology_is_Open_Extension_Topology_of_Discrete_Topology | [
"Open Extension Topologies",
"Excluded Point Topologies",
"Discrete Topologies"
] | [
"Definition:Set",
"Definition:Excluded Point Topology",
"Definition:Discrete Topology",
"Definition:Open Extension Topology"
] | [
"Definition:Excluded Point Topology",
"Definition:Discrete Topology",
"Definition:Open Extension Topology"
] |
proofwiki-3778 | Product of Countable Discrete Space with Sierpiński Space is Paracompact | Let $T_X = \struct {S, \tau}$ be a countable discrete space.
Let $T_Y = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.
Let $T_X \times T_Y$ be the product space of $T_X$ and $T_Y$.
Then $T_X \times T_Y$ is paracompact. | From Discrete Space is Paracompact, $T_X$ is paracompact.
We have that the Sierpiński space $T_Y$ is a finite topological space.
From Finite Topological Space is Compact, $T_Y$ is a compact space.
{{finish|Steen and Seebach in Part $\text I$ chapter $3$ Compactness: Invariance Properties offer "If $X$ is compact, then ... | Let $T_X = \struct {S, \tau}$ be a [[Definition:Countable Discrete Space|countable discrete space]].
Let $T_Y = \struct {\set {0, 1}, \tau_0}$ be a [[Definition:Sierpiński Space|Sierpiński space]].
Let $T_X \times T_Y$ be the [[Definition:Product Space|product space]] of $T_X$ and $T_Y$.
Then $T_X \times T_Y$ is [[... | From [[Discrete Space is Paracompact]], $T_X$ is [[Definition:Paracompact Space|paracompact]].
We have that the [[Definition:Sierpiński Space|Sierpiński space]] $T_Y$ is a [[Definition:Finite Topological Space|finite topological space]].
From [[Finite Topological Space is Compact]], $T_Y$ is a [[Definition:Compact To... | Product of Countable Discrete Space with Sierpiński Space is Paracompact | https://proofwiki.org/wiki/Product_of_Countable_Discrete_Space_with_Sierpiński_Space_is_Paracompact | https://proofwiki.org/wiki/Product_of_Countable_Discrete_Space_with_Sierpiński_Space_is_Paracompact | [
"Sierpiński Space",
"Discrete Topologies",
"Examples of Paracompact Spaces"
] | [
"Definition:Discrete Topology/Countable",
"Definition:Sierpiński Space",
"Definition:Product Space",
"Definition:Paracompact Space"
] | [
"Discrete Space is Paracompact",
"Definition:Paracompact Space",
"Definition:Sierpiński Space",
"Definition:Finite Topological Space",
"Finite Topological Space is Compact",
"Definition:Compact Topological Space"
] |
proofwiki-3779 | Reduction of Explicit ODE to First Order System | Let $\map {x^{\paren n} } t = \map F {t, x, x', \ldots, x^{\paren {n - 1} } }$, $\map x {t_0} = x_0$ be an explicit ODE with $x \in \R^m$.
Let there exist $I \subseteq \R$ such that there exists a unique particular solution:
:$x: I \to \R^m$
to this ODE.
Then there exists a system of first order ODEs:
:$y' = \map {\til... | Define the mappings:
:$z_1, \ldots, z_n: I \to \R^m$
by:
:$z_j = x^{\paren {j - 1} }$, $j = 1, \ldots, n$
Then:
{{begin-eqn}}
{{eqn | l = z_1'
| r = z_2
}}
{{eqn | o = \vdots
}}
{{eqn | l = z_{n - 1}'
| r = z_n
}}
{{eqn | l = z_n'
| r = \map F {t, z_1, \ldots, z_n}
}}
{{end-eqn}}
This is a system of $... | Let $\map {x^{\paren n} } t = \map F {t, x, x', \ldots, x^{\paren {n - 1} } }$, $\map x {t_0} = x_0$ be an [[Definition:Explicit ODE|explicit ODE]] with $x \in \R^m$.
Let there exist $I \subseteq \R$ such that there exists a unique [[Definition:Particular Solution of Differential Equation|particular solution]]:
:$x: I... | Define the [[Definition:Mapping|mappings]]:
:$z_1, \ldots, z_n: I \to \R^m$
by:
:$z_j = x^{\paren {j - 1} }$, $j = 1, \ldots, n$
Then:
{{begin-eqn}}
{{eqn | l = z_1'
| r = z_2
}}
{{eqn | o = \vdots
}}
{{eqn | l = z_{n - 1}'
| r = z_n
}}
{{eqn | l = z_n'
| r = \map F {t, z_1, \ldots, z_n}
}}
{{end-eqn... | Reduction of Explicit ODE to First Order System | https://proofwiki.org/wiki/Reduction_of_Explicit_ODE_to_First_Order_System | https://proofwiki.org/wiki/Reduction_of_Explicit_ODE_to_First_Order_System | [
"Ordinary Differential Equations"
] | [
"Definition:Differential Equation/Explicit",
"Definition:Differential Equation/Solution/Particular Solution",
"Definition:Differential Equation/Explicit",
"Definition:Differential Equation/System",
"Definition:First Order Ordinary Differential Equation"
] | [
"Definition:Mapping",
"Definition:Differential Equation/System",
"Definition:First Order Ordinary Differential Equation"
] |
proofwiki-3780 | Open Continuous Image of Paracompact Space is not always Countably Metacompact | Let $T_A = \struct {X_A, \tau_A}$ be a topological space which is paracompact.
Let $T_B = \struct {X_B, \tau_B}$ be another topological space.
Let $\phi: T_A \to T_B$ be a mapping which is both continuous and open.
Then it is not necessarily even the case that $T_B$ is countably metacompact, let alone paracompact. | Let $T_X = \struct {X, \tau}$ be a countable discrete space.
Let $T_Y = \struct {\set {0, 1}, \tau_0}$ be a Sierpiński space.
Let $T_A = T_X \times T_Y$ be the product space of $T_X$ and $T_Y$ whose product topology is $\tau_A$
Then $T_A$ is paracompact from Product of Countable Discrete Space with Sierpiński Space is ... | Let $T_A = \struct {X_A, \tau_A}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Paracompact Space|paracompact]].
Let $T_B = \struct {X_B, \tau_B}$ be another [[Definition:Topological Space|topological space]].
Let $\phi: T_A \to T_B$ be a [[Definition:Mapping|mapping]] which is both [[... | Let $T_X = \struct {X, \tau}$ be a [[Definition:Countable Discrete Topology|countable discrete space]].
Let $T_Y = \struct {\set {0, 1}, \tau_0}$ be a [[Definition:Sierpiński Space|Sierpiński space]].
Let $T_A = T_X \times T_Y$ be the [[Definition:Product Space (Topology) of Two Factor Spaces|product space]] of $T_X$... | Open Continuous Image of Paracompact Space is not always Countably Metacompact | https://proofwiki.org/wiki/Open_Continuous_Image_of_Paracompact_Space_is_not_always_Countably_Metacompact | https://proofwiki.org/wiki/Open_Continuous_Image_of_Paracompact_Space_is_not_always_Countably_Metacompact | [
"Paracompact Spaces",
"Countably Metacompact Spaces",
"Continuous Mappings (Topology)",
"Open Mappings"
] | [
"Definition:Topological Space",
"Definition:Paracompact Space",
"Definition:Topological Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Open Mapping",
"Definition:Countably Metacompact Space",
"Definition:Paracompact Space"
] | [
"Definition:Discrete Topology/Countable",
"Definition:Sierpiński Space",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Product Topology/Two Factor Spaces",
"Definition:Paracompact Space",
"Product of Countable Discrete Space with Sierpiński Space is Paracompact",
"Definition:Parti... |
proofwiki-3781 | Fixed Point Formulation of Explicit ODE | Let $x' = \map f {t, x}$ with $\map x {t_0} = x_0$ be an explicit ODE of dimension $n$.
For $a, b \in \R$, let $\XX = \map {\CC} {\closedint a b; \R^n}$ be the space of continuous functions on the closed interval $\closedint a b$.
Let $T: \XX \to \XX$ be the map defined by:
:$\ds \map {\paren {T x} } t = x_0 + \int_{t_... | Let $\map y t$ be a fixed point of the map $T$.
That is:
:$\ds \map y t = x_0 + \int_{t_0}^t \map f {s, \map y s} \rd s$
Then:
:$\ds \map y {t_0} = x_0 + \int_{t_0}^{t_0} \map f {s, \map y s} \rd s = x_0$
By the fundamental theorem of calculus we have that $y$ is differentiable, and for $t \in \closedint a b$:
{{begin-... | Let $x' = \map f {t, x}$ with $\map x {t_0} = x_0$ be an [[Definition:Explicit ODE|explicit ODE]] of [[Definition:Dimension of Differential Equation|dimension]] $n$.
For $a, b \in \R$, let $\XX = \map {\CC} {\closedint a b; \R^n}$ be the [[Continuous Functions on Interval Form Banach Space|space of continuous function... | Let $\map y t$ be a fixed point of the map $T$.
That is:
:$\ds \map y t = x_0 + \int_{t_0}^t \map f {s, \map y s} \rd s$
Then:
:$\ds \map y {t_0} = x_0 + \int_{t_0}^{t_0} \map f {s, \map y s} \rd s = x_0$
By the [[Fundamental Theorem of Calculus/First Part|fundamental theorem of calculus]] we have that $y$ is [[Defi... | Fixed Point Formulation of Explicit ODE | https://proofwiki.org/wiki/Fixed_Point_Formulation_of_Explicit_ODE | https://proofwiki.org/wiki/Fixed_Point_Formulation_of_Explicit_ODE | [
"Ordinary Differential Equations"
] | [
"Definition:Differential Equation/Explicit",
"Definition:Differential Equation/Order",
"Continuous Functions on Interval Form Banach Space",
"Definition:Real Interval/Closed",
"Definition:Mapping",
"Definition:Fixed Point",
"Definition:Differential Equation/Solution",
"Definition:Differential Equation... | [
"Fundamental Theorem of Calculus/First Part",
"Definition:Differentiable Mapping",
"Derivative of Constant",
"Fundamental Theorem of Calculus/First Part",
"Definition:Differential Equation/Ordinary",
"Category:Ordinary Differential Equations"
] |
proofwiki-3782 | Cardinality of Infinite Union of Infinite Sets | Let $\kappa$ be an infinite cardinal.
Let $X_i$ be sets of cardinality at most $\kappa$ indexed by a set $I$ of cardinality at most $\kappa$.
Then their union $\ds \bigcup_{i \mathop \in I} X_i$ has cardinality at most $\kappa$.
Furthermore, if at least one of the $X_i$ is size $\kappa$, then the union has cardinality ... | We can assume that $I$ has cardinality $\kappa$ and that the sets $X_i$ are disjoint and all of size $\kappa$.
The more general case follows since these other possibilities can only decrease the cardinality of the union.
When at least one of the $X_i$ is size $\kappa$, then the union must be at least size $\kappa$ sinc... | Let $\kappa$ be an [[Definition:Infinite|infinite]] [[Definition:Cardinal|cardinal]].
Let $X_i$ be [[Definition:Set|sets]] of [[Definition:Cardinality|cardinality]] at most $\kappa$ [[Definition:Indexing Set|indexed]] by a set $I$ of [[Definition:Cardinality|cardinality]] at most $\kappa$.
Then [[Definition:Set Union... | We can assume that $I$ has [[Definition:Cardinality|cardinality]] $\kappa$ and that the sets $X_i$ are disjoint and all of size $\kappa$.
The more general case follows since these other possibilities can only decrease the [[Definition:Cardinality|cardinality]] of the union.
When at least one of the $X_i$ is size $\ka... | Cardinality of Infinite Union of Infinite Sets | https://proofwiki.org/wiki/Cardinality_of_Infinite_Union_of_Infinite_Sets | https://proofwiki.org/wiki/Cardinality_of_Infinite_Union_of_Infinite_Sets | [
"Infinite Sets"
] | [
"Definition:Infinite",
"Definition:Cardinal",
"Definition:Set",
"Definition:Cardinality",
"Definition:Indexing Set",
"Definition:Cardinality",
"Definition:Set Union",
"Definition:Cardinality",
"Definition:Set Union",
"Definition:Cardinality"
] | [
"Definition:Cardinality",
"Definition:Cardinality",
"Definition:Limit Ordinal",
"Definition:Cardinality",
"Definition:Lexicographic Order"
] |
proofwiki-3783 | Zermelo's Well-Ordering Theorem | Let the Axiom of Choice be accepted.
Then every set is well-orderable. | Let $S$ be an arbitrary set.
By assumption $S$ is well-orderable.
From Well-Orderable Set has Choice Function, $S$ has a choice function.
As $S$ is arbitrary, the result follows.
{{qed}} | Let the [[Axiom:Axiom of Choice|Axiom of Choice]] be accepted.
Then every [[Definition:Set|set]] is [[Definition:Well-Orderable Set|well-orderable]]. | Let $S$ be an arbitrary [[Definition:Set|set]].
By assumption $S$ is [[Definition:Well-Orderable Set|well-orderable]].
From [[Well-Orderable Set has Choice Function]], $S$ has a [[Definition:Choice Function|choice function]].
As $S$ is arbitrary, the result follows.
{{qed}} | Zermelo's Well-Ordering Theorem/Converse/Proof 1 | https://proofwiki.org/wiki/Zermelo's_Well-Ordering_Theorem | https://proofwiki.org/wiki/Zermelo's_Well-Ordering_Theorem/Converse/Proof_1 | [
"Zermelo's Well-Ordering Theorem",
"Well-Orderings",
"Equivalents of Axiom of Choice"
] | [
"Axiom:Axiom of Choice",
"Definition:Set",
"Definition:Well-Orderable Set"
] | [
"Definition:Set",
"Definition:Well-Orderable Set",
"Well-Orderable Set has Choice Function",
"Definition:Choice Function"
] |
proofwiki-3784 | Zermelo's Well-Ordering Theorem | Let the Axiom of Choice be accepted.
Then every set is well-orderable. | Let $\FF$ be an arbitrary collection of sets.
By assumption all sets can be well-ordered.
Hence the set $\bigcup \FF$ of all elements of sets contained in $\FF$ is well-ordered by some ordering $\preceq$.
By definition then, every subset of $\ds \bigcup \FF$ has a smallest element under $\preceq$.
Also, note that each ... | Let the [[Axiom:Axiom of Choice|Axiom of Choice]] be accepted.
Then every [[Definition:Set|set]] is [[Definition:Well-Orderable Set|well-orderable]]. | Let $\FF$ be an arbitrary [[Definition:Set of Sets|collection of sets]].
By assumption all [[Definition:Set|sets]] can be [[Definition:Well-Ordered Set|well-ordered]].
Hence the [[Definition:Set|set]] $\bigcup \FF$ of all [[Definition:Element|elements]] of [[Definition:Set|sets]] contained in $\FF$ is [[Definition:W... | Zermelo's Well-Ordering Theorem/Converse/Proof 2 | https://proofwiki.org/wiki/Zermelo's_Well-Ordering_Theorem | https://proofwiki.org/wiki/Zermelo's_Well-Ordering_Theorem/Converse/Proof_2 | [
"Zermelo's Well-Ordering Theorem",
"Well-Orderings",
"Equivalents of Axiom of Choice"
] | [
"Axiom:Axiom of Choice",
"Definition:Set",
"Definition:Well-Orderable Set"
] | [
"Definition:Set of Sets",
"Definition:Set",
"Definition:Well-Ordered Set",
"Definition:Set",
"Definition:Element",
"Definition:Set",
"Definition:Well-Ordered Set",
"Definition:Ordering",
"Definition:Subset",
"Definition:Smallest Element",
"Definition:Set",
"Definition:Subset",
"Definition:Ch... |
proofwiki-3785 | Zermelo's Well-Ordering Theorem | Let the Axiom of Choice be accepted.
Then every set is well-orderable. | Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
By the {{Axiom-link|Choice}}, there exists a choice function on $S$.
The result follows from Set with Choice Function is Well-Orderable.
{{qed}} | Let the [[Axiom:Axiom of Choice|Axiom of Choice]] be accepted.
Then every [[Definition:Set|set]] is [[Definition:Well-Orderable Set|well-orderable]]. | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ be the [[Definition:Power Set|power set]] of $S$.
By the {{Axiom-link|Choice}}, there exists a [[Definition:Choice Function|choice function]] on $S$.
The result follows from [[Set with Choice Function is Well-Orderable]].
{{qed}} | Zermelo's Well-Ordering Theorem/Proof 1 | https://proofwiki.org/wiki/Zermelo's_Well-Ordering_Theorem | https://proofwiki.org/wiki/Zermelo's_Well-Ordering_Theorem/Proof_1 | [
"Zermelo's Well-Ordering Theorem",
"Well-Orderings",
"Equivalents of Axiom of Choice"
] | [
"Axiom:Axiom of Choice",
"Definition:Set",
"Definition:Well-Orderable Set"
] | [
"Definition:Set",
"Definition:Power Set",
"Definition:Choice Function",
"Set with Choice Function is Well-Orderable"
] |
proofwiki-3786 | Zermelo's Well-Ordering Theorem | Let the Axiom of Choice be accepted.
Then every set is well-orderable. | Let $S$ be a set.
Let $\powerset S$ be the power set of $S$.
By the Axiom of Choice, there is a choice function $c$ defined on $\powerset S \setminus \set \O$.
We will use $c$ and the Principle of Transfinite Induction to define a bijection between $S$ and some ordinal.
Intuitively, we start by pairing $\map c S$ with ... | Let the [[Axiom:Axiom of Choice|Axiom of Choice]] be accepted.
Then every [[Definition:Set|set]] is [[Definition:Well-Orderable Set|well-orderable]]. | Let $S$ be a [[Definition:Set|set]].
Let $\powerset S$ be the [[Definition:Power Set|power set]] of $S$.
By the [[Axiom:Axiom of Choice|Axiom of Choice]], there is a [[Definition:Choice Function|choice function]] $c$ defined on $\powerset S \setminus \set \O$.
We will use $c$ and the [[Principle of Transfinite Induc... | Zermelo's Well-Ordering Theorem/Proof 2 | https://proofwiki.org/wiki/Zermelo's_Well-Ordering_Theorem | https://proofwiki.org/wiki/Zermelo's_Well-Ordering_Theorem/Proof_2 | [
"Zermelo's Well-Ordering Theorem",
"Well-Orderings",
"Equivalents of Axiom of Choice"
] | [
"Axiom:Axiom of Choice",
"Definition:Set",
"Definition:Well-Orderable Set"
] | [
"Definition:Set",
"Definition:Power Set",
"Axiom:Axiom of Choice",
"Definition:Choice Function",
"Transfinite Induction",
"Definition:Bijection",
"Definition:Ordinal",
"Definition:Bijection",
"Definition:Empty Set",
"Hartogs' Lemma (Set Theory)",
"Definition:Well-Ordering",
"Definition:Well-Or... |
proofwiki-3787 | Zermelo's Well-Ordering Theorem | Let the Axiom of Choice be accepted.
Then every set is well-orderable. | Let $S$ be a non-empty set.
Let $C$ be a choice function for $S$.
Let $A$ be an arbitrary set.
Let the mapping $h$ be defined as:
:$\map h A = \begin {cases} \map C {S \setminus A} & : A \subsetneqq S \\ x & : \text {otherwise} \end {cases}$
where $x$ is an arbitrary element such that $x \notin S$.
The latter is known ... | Let the [[Axiom:Axiom of Choice|Axiom of Choice]] be accepted.
Then every [[Definition:Set|set]] is [[Definition:Well-Orderable Set|well-orderable]]. | Let $S$ be a [[Definition:Non-Empty Set|non-empty set]].
Let $C$ be a [[Definition:Choice Function|choice function]] for $S$.
Let $A$ be an arbitrary [[Definition:Set|set]].
Let the [[Definition:Mapping|mapping]] $h$ be defined as:
:$\map h A = \begin {cases} \map C {S \setminus A} & : A \subsetneqq S \\ x & : \tex... | Zermelo's Well-Ordering Theorem/Proof 3 | https://proofwiki.org/wiki/Zermelo's_Well-Ordering_Theorem | https://proofwiki.org/wiki/Zermelo's_Well-Ordering_Theorem/Proof_3 | [
"Zermelo's Well-Ordering Theorem",
"Well-Orderings",
"Equivalents of Axiom of Choice"
] | [
"Axiom:Axiom of Choice",
"Definition:Set",
"Definition:Well-Orderable Set"
] | [
"Definition:Non-Empty Set",
"Definition:Choice Function",
"Definition:Set",
"Definition:Mapping",
"Definition:Element",
"Exists Element Not in Set",
"Transfinite Recursion Theorem/Formulation 5",
"Definition:Mapping/Class Theory",
"Definition:Class of All Ordinals",
"Definition:Ordinal",
"Defini... |
proofwiki-3788 | Zermelo's Well-Ordering Theorem is Equivalent to Axiom of Choice | Zermelo's Well-Ordering Theorem holds {{iff}} the Axiom of Choice holds.
That is, every set is well-orderable {{iff}} every collection of sets has a choice function. | === Necessary Condition ===
Suppose the Axiom of Choice holds.
Then Zermelo's Well-Ordering Theorem holds by Zermelo's Well-Ordering Theorem itself.
That is, every set is well-orderable.
{{qed|lemma}} | [[Zermelo's Well-Ordering Theorem]] holds {{iff}} the [[Axiom:Axiom of Choice|Axiom of Choice]] holds.
That is, every [[Definition:Set|set]] is [[Definition:Well-Orderable Class|well-orderable]] {{iff}} every [[Definition:Set of Sets|collection of sets]] has a [[Definition:Choice Function|choice function]]. | === Necessary Condition ===
Suppose the [[Axiom:Axiom of Choice|Axiom of Choice]] holds.
Then [[Zermelo's Well-Ordering Theorem]] holds by [[Zermelo's Well-Ordering Theorem]] itself.
That is, every [[Definition:Set|set]] is [[Definition:Well-Orderable Class|well-orderable]].
{{qed|lemma}} | Zermelo's Well-Ordering Theorem is Equivalent to Axiom of Choice | https://proofwiki.org/wiki/Zermelo's_Well-Ordering_Theorem_is_Equivalent_to_Axiom_of_Choice | https://proofwiki.org/wiki/Zermelo's_Well-Ordering_Theorem_is_Equivalent_to_Axiom_of_Choice | [
"Axiom of Choice",
"Zermelo's Well-Ordering Theorem"
] | [
"Zermelo's Well-Ordering Theorem",
"Axiom:Axiom of Choice",
"Definition:Set",
"Definition:Well-Orderable Set/Class Theory",
"Definition:Set of Sets",
"Definition:Choice Function"
] | [
"Axiom:Axiom of Choice",
"Zermelo's Well-Ordering Theorem",
"Zermelo's Well-Ordering Theorem",
"Definition:Set",
"Definition:Well-Orderable Set/Class Theory",
"Definition:Set",
"Axiom:Axiom of Choice"
] |
proofwiki-3789 | Product Topology is Coarsest Topology such that Projections are Continuous | Let $\mathbb X = \family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an indexed family of topological spaces where $I$ is an arbitrary index set.
Let $X$ be the cartesian product of $\mathbb X$:
:$\ds X := \prod_{i \mathop \in I} X_i$
Let $\tau$ be the product topology on $X$.
For each $i \in I$, let $\pr_i : X \to ... | The result follows from the definition of the product topology and Equivalence of Definitions of Initial Topology.
{{qed}} | Let $\mathbb X = \family {\struct {X_i, \tau_i} }_{i \mathop \in I}$ be an [[Definition:Indexed Family|indexed family]] of [[Definition:Topological Space|topological spaces]] where $I$ is an arbitrary [[Definition:Indexing Set|index set]].
Let $X$ be the [[Definition:Cartesian Product|cartesian product]] of $\mathbb X... | The result follows from the definition of the [[Definition:Product Topology|product topology]] and [[Equivalence of Definitions of Initial Topology]].
{{qed}} | Product Topology is Coarsest Topology such that Projections are Continuous | https://proofwiki.org/wiki/Product_Topology_is_Coarsest_Topology_such_that_Projections_are_Continuous | https://proofwiki.org/wiki/Product_Topology_is_Coarsest_Topology_such_that_Projections_are_Continuous | [
"Product Topology",
"Projections",
"Coarser Topology"
] | [
"Definition:Indexing Set/Family",
"Definition:Topological Space",
"Definition:Indexing Set",
"Definition:Cartesian Product",
"Definition:Product Topology",
"Definition:Projection (Mapping Theory)",
"Definition:Ordered Tuple",
"Definition:Coarser Topology",
"Definition:Continuous Mapping (Topology)"
... | [
"Definition:Product Topology",
"Equivalence of Definitions of Initial Topology"
] |
proofwiki-3790 | Directed Hamilton Cycle Problem is NP-complete | Both versions of the Directed Hamilton Cycle problem are NP-complete. | === Function Version is Reducible to Decision Version ===
The algorithm described below solves the function version of the problem with $O \left({n^2}\right)$ calls of the decision version of the problem.
;Input: The directed graph $G$
;Output: Either:
:A Hamilton cycle in $G$ if one exists
or:
:''no solution'' if not.... | Both versions of the [[Definition:Directed Hamilton Cycle Problem|Directed Hamilton Cycle problem]] are [[Definition:NP-Complete|NP-complete]]. | === Function Version is Reducible to Decision Version ===
The algorithm described below solves the [[Definition:Directed Hamilton Cycle Problem/Function Version|function version]] of the problem with $O \left({n^2}\right)$ calls of the [[Definition:Directed Hamilton Cycle Problem/Decision Version|decision version]] of... | Directed Hamilton Cycle Problem is NP-complete | https://proofwiki.org/wiki/Directed_Hamilton_Cycle_Problem_is_NP-complete | https://proofwiki.org/wiki/Directed_Hamilton_Cycle_Problem_is_NP-complete | [
"Hamiltonian Graphs",
"Computer Science",
"Open Questions"
] | [
"Definition:Directed Hamilton Cycle Problem",
"Definition:NP-Complete"
] | [
"Definition:Directed Hamilton Cycle Problem/Function Version",
"Definition:Directed Hamilton Cycle Problem/Decision Version",
"Definition:Digraph",
"Definition:Hamilton Cycle",
"Definition:Hamilton Cycle",
"Definition:Hamilton Cycle",
"Definition:Graph (Graph Theory)/Vertex",
"Definition:Directed Hami... |
proofwiki-3791 | Condition for Closed Extension Space to be T0 Space | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the closed extension space of $T$.
Then $T^*_p$ is a $T_0$ space {{iff}} $T$ is. | By definition:
:$\tau^*_p = \set {U \cup \set p: U \in \tau} \cup \set \O$
Let $T = \struct {S, \tau}$ be a $T_0$ space.
Then:
:$\forall x, y \in S$ such that $x \ne y$, either:
::$\exists U \in \tau: x \in U, y \notin U$
:or:
::$\exists U \in \tau: y \in U, x \notin U$
Let $x, y \in S^*_p$ such that $x \ne y, x \ne p,... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_p = \struct {S^*_p, \tau^*_p}$ be the [[Definition:Closed Extension Space|closed extension space]] of $T$.
Then $T^*_p$ is a [[Definition:T0 Space|$T_0$ space]] {{iff}} $T$ is. | By definition:
:$\tau^*_p = \set {U \cup \set p: U \in \tau} \cup \set \O$
Let $T = \struct {S, \tau}$ be a [[Definition:T0 Space|$T_0$ space]].
Then:
:$\forall x, y \in S$ such that $x \ne y$, either:
::$\exists U \in \tau: x \in U, y \notin U$
:or:
::$\exists U \in \tau: y \in U, x \notin U$
Let $x, y \in S^*_... | Condition for Closed Extension Space to be T0 Space | https://proofwiki.org/wiki/Condition_for_Closed_Extension_Space_to_be_T0_Space | https://proofwiki.org/wiki/Condition_for_Closed_Extension_Space_to_be_T0_Space | [
"Closed Extension Topologies",
"Examples of T0 Spaces"
] | [
"Definition:Topological Space",
"Definition:Closed Extension Topology",
"Definition:T0 Space"
] | [
"Definition:T0 Space",
"Definition:T0 Space",
"Definition:T0 Space",
"Definition:T0 Space"
] |
proofwiki-3792 | Excluded Point Space is T0 | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is a $T_0$ space. | Let $T$ be a trivial space.
That is, let $S = \set p$.
Then the result holds vacuously, as there are no two distinct points in $T$.
Now suppose $T$ is not trivial.
Then $\exists x \in S: x \ne p$.
Now we have that $\set x \subseteq T$ is open in $T$ such that $p \notin \set x$ but $x \in \set x$.
Finally, suppose that ... | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Space|excluded point space]].
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | Let $T$ be a [[Definition:Trivial Topological Space|trivial space]].
That is, let $S = \set p$.
Then the result holds [[Definition:Vacuous Truth|vacuously]], as there are no two [[Definition:Distinct Elements|distinct points]] in $T$.
Now suppose $T$ is not [[Definition:Trivial Topological Space|trivial]].
Then $\... | Excluded Point Space is T0/Proof 1 | https://proofwiki.org/wiki/Excluded_Point_Space_is_T0 | https://proofwiki.org/wiki/Excluded_Point_Space_is_T0/Proof_1 | [
"Excluded Point Space is T0",
"Excluded Point Topologies",
"Examples of T0 Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:T0 Space"
] | [
"Definition:Trivial Topological Space",
"Definition:Vacuous Truth",
"Definition:Distinct/Plural",
"Definition:Trivial Topological Space",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology"
] |
proofwiki-3793 | Excluded Point Space is T0 | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space.
Then $T$ is a $T_0$ space. | We have:
:Excluded Point Topology is Open Extension Topology of Discrete Topology
:Discrete Space is $T_0$
Then by Condition for Open Extension Space to be $T_0$, as a discrete space is $T_0$ then so is its open extension.
{{qed}} | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Space|excluded point space]].
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | We have:
:[[Excluded Point Topology is Open Extension Topology of Discrete Topology]]
:[[Discrete Space is T0|Discrete Space is $T_0$]]
Then by [[Condition for Open Extension Space to be T0|Condition for Open Extension Space to be $T_0$]], as a [[Definition:Discrete Space|discrete space]] is $T_0$ then so is its [[D... | Excluded Point Space is T0/Proof 2 | https://proofwiki.org/wiki/Excluded_Point_Space_is_T0 | https://proofwiki.org/wiki/Excluded_Point_Space_is_T0/Proof_2 | [
"Excluded Point Space is T0",
"Excluded Point Topologies",
"Examples of T0 Spaces"
] | [
"Definition:Excluded Point Topology",
"Definition:T0 Space"
] | [
"Excluded Point Topology is Open Extension Topology of Discrete Topology",
"Discrete Space is T0",
"Condition for Open Extension Space to be T0",
"Definition:Discrete Topology",
"Definition:Open Extension Topology"
] |
proofwiki-3794 | Open Extension Topology is not T1 | Let $T = \struct {S, \tau}$ be a topological space.
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the open extension space of $T$.
Then $T^*_{\bar p}$ is not a $T_1$ space. | By definition:
:$\tau^*_{\bar p} = \set {U: U \in \tau} \cup \set {S^*_p}$
Let $x \in S^*_p, x \ne p$.
Let $U = \set x$.
Then $U \in \tau^*_p$ such that $x \in U, p \notin U$.
But the only $v \in \tau^*_p$ such that $p \in V$ is the set $S^*_p$, and we have that $x \in S^*_p$.
So there is no $V \in \tau^*_p$ such that ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Let $T^*_{\bar p} = \struct {S^*_p, \tau^*_{\bar p} }$ be the [[Definition:Open Extension Space|open extension space]] of $T$.
Then $T^*_{\bar p}$ is not a [[Definition:T1 Space|$T_1$ space]]. | By definition:
:$\tau^*_{\bar p} = \set {U: U \in \tau} \cup \set {S^*_p}$
Let $x \in S^*_p, x \ne p$.
Let $U = \set x$.
Then $U \in \tau^*_p$ such that $x \in U, p \notin U$.
But the only $v \in \tau^*_p$ such that $p \in V$ is the set $S^*_p$, and we have that $x \in S^*_p$.
So there is no $V \in \tau^*_p$ suc... | Open Extension Topology is not T1 | https://proofwiki.org/wiki/Open_Extension_Topology_is_not_T1 | https://proofwiki.org/wiki/Open_Extension_Topology_is_not_T1 | [
"Open Extension Topologies",
"Examples of T1 Spaces"
] | [
"Definition:Topological Space",
"Definition:Open Extension Topology",
"Definition:T1 Space"
] | [
"Definition:Open Extension Topology",
"Definition:T1 Space"
] |
proofwiki-3795 | Extreme Value Theorem | Let $X$ be a compact metric space and $Y$ a normed vector space.
Let $f: X \to Y$ be a continuous mapping.
Then $f$ is bounded, and there exist $x, y \in X$ such that:
:$\forall z \in X: \norm {\map f x} \le \norm {\map f z} \le \norm {\map f y}$
where $\norm {\map f x}$ denotes the norm of $\map f x$.
Moreover, $\norm... | {{AxiomReview|ZF version of this proof (use Continuous Image of Compact Space is Compact/Corollary 3)}}
{{explain|What should be reviewed??}}
By Continuous Image of Compact Space is Compact, $f \sqbrk X \subseteq Y$ is compact.
Therefore, by Compact Subspace of Metric Space is Bounded, $f$ is bounded.
Let $\ds A = \inf... | Let $X$ be a [[Definition:Compact Metric Space|compact metric space]] and $Y$ a [[Definition:Normed Vector Space|normed vector space]].
Let $f: X \to Y$ be a [[Definition:Continuous Mapping (Metric Spaces)|continuous mapping]].
Then $f$ is [[Definition:Bounded Mapping to Metric Space|bounded]], and there exist $x, y... | {{AxiomReview|[[Definition:Zermelo-Fraenkel Set Theory|ZF]] version of this proof (use [[Continuous Image of Compact Space is Compact/Corollary 3]])}}
{{explain|What should be reviewed??}}
By [[Continuous Image of Compact Space is Compact]], $f \sqbrk X \subseteq Y$ is [[Definition:Compact Metric Space|compact]].
The... | Extreme Value Theorem | https://proofwiki.org/wiki/Extreme_Value_Theorem | https://proofwiki.org/wiki/Extreme_Value_Theorem | [
"Metric Spaces",
"Continuous Functions"
] | [
"Definition:Compact Space/Metric Space",
"Definition:Normed Vector Space",
"Definition:Continuous Mapping (Metric Space)",
"Definition:Bounded Mapping/Metric Space",
"Definition:Norm/Vector Space",
"Definition:Minimum Value of Real Function/Absolute",
"Definition:Maximum Value of Real Function/Absolute"... | [
"Definition:Zermelo-Fraenkel Set Theory",
"Continuous Image of Compact Space is Compact/Corollary 3",
"Continuous Image of Compact Space is Compact",
"Definition:Compact Space/Metric Space",
"Compact Subspace of Metric Space is Bounded",
"Definition:Bounded Mapping/Metric Space",
"Definition:Infimum of ... |
proofwiki-3796 | Continuous Functions on Compact Space form Banach Space | Let $X$ be a compact Hausdorff space.
Let $Y$ be a Banach space.
Let $\CC = \CC \struct {X; Y}$ be the space of continuous functions on $X$ valued in $Y$.
Let $\norm {\,\cdot\,}_\infty$ be the supremum norm on $\CC$.
Then $\struct {\CC, \norm {\,\cdot\,}_\infty}$ is a Banach space. | From Continuous Function on Compact Space is Bounded, every continuous mapping $f : X \to Y$ is bounded.
Hence $\map \CC {X; Y} = \map {\CC_b} {X; Y}$, where $\map {\CC_b} {X; Y}$ is the space of bounded continuous functions on $X$ valued in $Y$.
From Bounded Continuous Functions on Topological Space form Banach Space,... | Let $X$ be a [[Definition:Compact Topological Subspace|compact]] [[Definition:Hausdorff Space|Hausdorff space]].
Let $Y$ be a [[Definition:Banach Space|Banach space]].
Let $\CC = \CC \struct {X; Y}$ be the [[Definition:Space of Continuous Functions on Compact Hausdorff Space|space of continuous functions on $X$ value... | From [[Continuous Function on Compact Space is Bounded]], every [[Definition:Continuous Mapping|continuous mapping]] $f : X \to Y$ is [[Definition:Bounded Mapping|bounded]].
Hence $\map \CC {X; Y} = \map {\CC_b} {X; Y}$, where $\map {\CC_b} {X; Y}$ is the [[Definition:Space of Bounded Continuous Functions on Topologic... | Continuous Functions on Compact Space form Banach Space | https://proofwiki.org/wiki/Continuous_Functions_on_Compact_Space_form_Banach_Space | https://proofwiki.org/wiki/Continuous_Functions_on_Compact_Space_form_Banach_Space | [
"Functional Analysis"
] | [
"Definition:Compact Topological Space/Subspace",
"Definition:T2 Space",
"Definition:Banach Space",
"Definition:Space of Continuous Functions on Compact Hausdorff Space",
"Definition:Supremum Norm",
"Definition:Banach Space"
] | [
"Continuous Function on Compact Space is Bounded",
"Definition:Continuous Mapping",
"Definition:Bounded Mapping",
"Definition:Space of Bounded Continuous Functions on Topological Space",
"Bounded Continuous Functions on Topological Space form Banach Space",
"Definition:Banach Space",
"Category:Functiona... |
proofwiki-3797 | Supremum Norm is Norm | Let $S$ be a set.
Let $\struct {X, \norm {\, \cdot \,} }$ be a normed vector space over $K \in \set {\R, \C}$.
Let $\BB$ be the set of bounded mappings $S \to X$.
Let $\norm {\, \cdot \,}_\infty$ be the supremum norm on $\BB$.
Then $\norm {\, \cdot \,}_\infty$ is a norm on $\BB$.
{{MissingLinks|Add a link that establis... | First:
{{begin-eqn}}
{{eqn | l = \norm f_\infty
| r = 0
}}
{{eqn | ll= \leadstoandfrom
| l = \sup_{x \mathop \in S} \norm {\map f x}
| r = 0
}}
{{eqn | ll= \leadstoandfrom
| q = \forall x \in S
| l = \norm {\map f x}
| r = 0
| c = since $\norm {\, \cdot \,}$ is a norm, and henc... | Let $S$ be a [[Definition:Set|set]].
Let $\struct {X, \norm {\, \cdot \,} }$ be a [[Definition:Normed Vector Space|normed vector space]] over $K \in \set {\R, \C}$.
Let $\BB$ be the set of [[Definition:Bounded Mapping|bounded mappings]] $S \to X$.
Let $\norm {\, \cdot \,}_\infty$ be the [[Definition:Supremum Norm|su... | First:
{{begin-eqn}}
{{eqn | l = \norm f_\infty
| r = 0
}}
{{eqn | ll= \leadstoandfrom
| l = \sup_{x \mathop \in S} \norm {\map f x}
| r = 0
}}
{{eqn | ll= \leadstoandfrom
| q = \forall x \in S
| l = \norm {\map f x}
| r = 0
| c = since $\norm {\, \cdot \,}$ is a [[Definition:... | Supremum Norm is Norm | https://proofwiki.org/wiki/Supremum_Norm_is_Norm | https://proofwiki.org/wiki/Supremum_Norm_is_Norm | [
"Examples of Norms",
"Supremum Norm",
"Supremum Norm is Norm"
] | [
"Definition:Set",
"Definition:Normed Vector Space",
"Definition:Bounded Mapping",
"Definition:Supremum Norm",
"Definition:Norm/Vector Space"
] | [
"Definition:Norm/Vector Space",
"Multiple of Supremum",
"Definition:Norm/Vector Space",
"Definition:Norm/Vector Space",
"Supremum of Sum",
"Definition:Norm/Vector Space",
"Category:Examples of Norms",
"Category:Supremum Norm",
"Category:Supremum Norm is Norm"
] |
proofwiki-3798 | One-Step Vector Subspace Test | Let $V$ be a vector space over a division ring $K$ whose unity is $1_K$.
Let $U \subseteq V$ be a non-empty subset of $V$ such that:
:$\forall u, v \in U: \forall \lambda \in K: u + \lambda v \in U$
Then $U$ is a subspace of $V$. | We start with observing that the properties:
:{{Vector-space-axiom|1}}
:{{Vector-space-axiom|2}}
:{{Vector-space-axiom|3}}
:{{Vector-space-axiom|4}}
are true for all elements of $V$.
Hence, since $U \subseteq V$, they hold for all elements of $U$ as well.
The same holds for the axioms:
:{{Abelian-group-axiom|C||underly... | Let $V$ be a [[Definition:Vector Space|vector space]] over a [[Definition:Division Ring|division ring]] $K$ whose [[Definition:Unity of Ring|unity]] is $1_K$.
Let $U \subseteq V$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $V$ such that:
:$\forall u, v \in U: \forall \lambda \in K: u +... | We start with observing that the properties:
:{{Vector-space-axiom|1}}
:{{Vector-space-axiom|2}}
:{{Vector-space-axiom|3}}
:{{Vector-space-axiom|4}}
are true for all [[Definition:Element|elements]] of $V$.
Hence, since $U \subseteq V$, they hold for all [[Definition:Element|elements]] of $U$ as well.
The same holds... | One-Step Vector Subspace Test | https://proofwiki.org/wiki/One-Step_Vector_Subspace_Test | https://proofwiki.org/wiki/One-Step_Vector_Subspace_Test | [
"Linear Algebra",
"Vector Subspaces"
] | [
"Definition:Vector Space",
"Definition:Division Ring",
"Definition:Unity (Abstract Algebra)/Ring",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Vector Subspace"
] | [
"Definition:Element",
"Definition:Element",
"Vector Inverse is Negative Vector",
"Definition:Element",
"Definition:Non-Empty Set",
"Axiom:Vector Space Axioms",
"Definition:Vector Subspace"
] |
proofwiki-3799 | Ax-Grothendieck Theorem | Let $f: \C^n \to \C^n$ be a polynomial map.
Let $f$ be injective.
Then $f$ is surjective. | The proof proceeds as follows:
:$(1): \quad$ showing that the theorem can be captured using a first-order sentences in the language of rings
:$(2): \quad$ showing that the theorem is true for at least one field of every characteristic $p > 0$
:$(3): \quad$ applying the Lefschetz Principle (First-Order). | Let $f: \C^n \to \C^n$ be a [[Definition:Polynomial Map|polynomial map]].
Let $f$ be [[Definition:Injection|injective]].
Then $f$ is [[Definition:Surjection|surjective]]. | The proof proceeds as follows:
:$(1): \quad$ showing that the theorem can be captured using a first-order sentences in the language of rings
:$(2): \quad$ showing that the theorem is true for at least one field of every characteristic $p > 0$
:$(3): \quad$ applying the [[Lefschetz Principle (First-Order)]]. | Ax-Grothendieck Theorem | https://proofwiki.org/wiki/Ax-Grothendieck_Theorem | https://proofwiki.org/wiki/Ax-Grothendieck_Theorem | [
"Model Theory for Predicate Logic",
"Complex Analysis"
] | [
"Definition:Polynomial Map",
"Definition:Injection",
"Definition:Surjection"
] | [
"Lefschetz Principle (First-Order)",
"Lefschetz Principle (First-Order)"
] |
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