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-3600 | Mapping from Discrete Space is Continuous | Let $T_1 = \struct {S_1, \tau_1}$ be the discrete topological space on $S_1$.
Let $T_2 = \struct {S_2, \tau_2}$ be another arbitrary topological space.
Let $\phi: S_1 \to S_2$ be a mapping.
Then $\phi$ is continuous. | From the definition of continuous:
:$U \in \tau_2 \implies \phi^{-1} \sqbrk U \in \tau_1$
But as $\phi^{-1} \sqbrk U \subseteq S_1$ it follows from the definition of discrete space that $\phi^{-1} \sqbrk U \in \tau_1$.
{{qed}} | Let $T_1 = \struct {S_1, \tau_1}$ be the [[Definition:Discrete Space|discrete topological space]] on $S_1$.
Let $T_2 = \struct {S_2, \tau_2}$ be another [[Definition:Arbitrary|arbitrary]] [[Definition:Topological Space|topological space]].
Let $\phi: S_1 \to S_2$ be a [[Definition:Mapping|mapping]].
Then $\phi$ is ... | From the definition of [[Definition:Everywhere Continuous Mapping (Topology)|continuous]]:
:$U \in \tau_2 \implies \phi^{-1} \sqbrk U \in \tau_1$
But as $\phi^{-1} \sqbrk U \subseteq S_1$ it follows from the definition of [[Definition:Discrete Space|discrete space]] that $\phi^{-1} \sqbrk U \in \tau_1$.
{{qed}} | Mapping from Discrete Space is Continuous | https://proofwiki.org/wiki/Mapping_from_Discrete_Space_is_Continuous | https://proofwiki.org/wiki/Mapping_from_Discrete_Space_is_Continuous | [
"Discrete Topologies",
"Continuous Mappings (Topology)"
] | [
"Definition:Discrete Topology",
"Definition:Arbitrary",
"Definition:Topological Space",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)/Everywhere"
] | [
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Discrete Topology"
] |
proofwiki-3601 | Exponential Function is Superfunction | The function $f : \C \to \C$, defined as:
:$\map f z = c^z$
is a superfunction for any complex number $c$. | Define $h: \C \to \C$ by $\map h z = z \times c$.
Then:
{{begin-eqn}}
{{eqn | l = \map h {\map f z}
| r = \map h {c^z}
| c =
}}
{{eqn | r = c^z \times c
| c =
}}
{{eqn | r = c^{z + 1}
| c =
}}
{{eqn | r = \map f {z + 1}
| c =
}}
{{end-eqn}}
Thus $\map f z = c^z$ is a superfunction and $\ma... | The [[Definition:Function|function]] $f : \C \to \C$, defined as:
:$\map f z = c^z$
is a [[Definition:Superfunction|superfunction]] for any [[Definition:Complex Number|complex number]] $c$. | Define $h: \C \to \C$ by $\map h z = z \times c$.
Then:
{{begin-eqn}}
{{eqn | l = \map h {\map f z}
| r = \map h {c^z}
| c =
}}
{{eqn | r = c^z \times c
| c =
}}
{{eqn | r = c^{z + 1}
| c =
}}
{{eqn | r = \map f {z + 1}
| c =
}}
{{end-eqn}}
Thus $\map f z = c^z$ is a superfunction and $\... | Exponential Function is Superfunction | https://proofwiki.org/wiki/Exponential_Function_is_Superfunction | https://proofwiki.org/wiki/Exponential_Function_is_Superfunction | [
"Superfunctions"
] | [
"Definition:Function",
"Definition:Superfunction",
"Definition:Complex Number"
] | [
"Definition:Superfunction",
"Category:Superfunctions"
] |
proofwiki-3602 | Standard Discrete Metric induces Discrete Topology | Let $M = \struct {A, d}$ be the (standard) discrete metric space on $A$.
Then $d$ induces the discrete topology on $A$.
Thus the discrete topology is metrizable. | Let $a \in A$.
From Subset of Standard Discrete Metric Space is Open, a set $U \subseteq A$ is open in $M$.
So, in particular, $\set a$ is open in $\struct {A, d}$.
This holds for all $a \in A$.
From Metric Induces Topology it follows that $\set a$ is an open set in $\struct {A, \tau_{A, d} }$.
The result follows from ... | Let $M = \struct {A, d}$ be the [[Definition:Standard Discrete Metric|(standard) discrete metric space]] on $A$.
Then [[Metric Induces Topology|$d$ induces]] the [[Definition:Discrete Topology|discrete topology]] on $A$.
Thus the [[Definition:Discrete Topology|discrete topology]] is [[Definition:Metrizable Space|me... | Let $a \in A$.
From [[Subset of Standard Discrete Metric Space is Open]], a set $U \subseteq A$ is [[Definition:Open Set (Metric Space)|open]] in $M$.
So, in particular, $\set a$ is [[Definition:Open Set (Metric Space)|open]] in $\struct {A, d}$.
This holds for all $a \in A$.
From [[Metric Induces Topology]] it fo... | Standard Discrete Metric induces Discrete Topology | https://proofwiki.org/wiki/Standard_Discrete_Metric_induces_Discrete_Topology | https://proofwiki.org/wiki/Standard_Discrete_Metric_induces_Discrete_Topology | [
"Standard Discrete Metric",
"Discrete Topologies",
"Examples of Metrizable Spaces"
] | [
"Definition:Standard Discrete Metric",
"Metric Induces Topology",
"Definition:Discrete Topology",
"Definition:Discrete Topology",
"Definition:Metrizable Space"
] | [
"Subset of Standard Discrete Metric Space is Open",
"Definition:Open Set/Metric Space",
"Definition:Open Set/Metric Space",
"Metric Induces Topology",
"Definition:Open Set/Topology",
"Basis for Discrete Topology"
] |
proofwiki-3603 | Discrete Space satisfies all Separation Properties | Let $T = \struct {S, \powerset S}$ be the discrete topological space on $S$.
Then $T$ fulfils all separation axioms. | We have that a Standard Discrete Metric induces Discrete Topology.
Then we use Metric Space fulfils all Separation Axioms.
{{qed}} | Let $T = \struct {S, \powerset S}$ be the [[Definition:Discrete Space|discrete topological space]] on $S$.
Then $T$ fulfils all [[Definition:Separation Axioms|separation axioms]]. | We have that a [[Standard Discrete Metric induces Discrete Topology]].
Then we use [[Metric Space fulfils all Separation Axioms]].
{{qed}} | Discrete Space satisfies all Separation Properties | https://proofwiki.org/wiki/Discrete_Space_satisfies_all_Separation_Properties | https://proofwiki.org/wiki/Discrete_Space_satisfies_all_Separation_Properties | [
"Discrete Space satisfies all Separation Properties",
"Discrete Topologies",
"Examples of Separation Axioms"
] | [
"Definition:Discrete Topology",
"Definition:Tychonoff Separation Axioms"
] | [
"Standard Discrete Metric induces Discrete Topology",
"Metric Space fulfils all Separation Axioms"
] |
proofwiki-3604 | Point in Discrete Space is Neighborhood | Let $T = \struct {S, \tau}$ be a discrete topological space.
Let $x \in S$.
Then $\set x$ is a neighborhood of $x$ in $T$. | By definition, a neighborhood $N_x$ of $x$ is any subset of $S$ containing an open set which itself contains $x$.
That is:
:$\exists U \in \tau: x \in U \subseteq N_x \subseteq S$
From Set in Discrete Topology is Clopen we have that $\set x$ is open set in $S$.
So by Set is Subset of Itself, $\set x$ is a subset of $S$... | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Topology|discrete topological space]].
Let $x \in S$.
Then $\set x$ is a [[Definition:Neighborhood of Point|neighborhood]] of $x$ in $T$. | By definition, a [[Definition:Neighborhood of Point|neighborhood]] $N_x$ of $x$ is any [[Definition:Subset|subset]] of $S$ containing an [[Definition:Open Set (Topology)|open set]] which itself contains $x$.
That is:
:$\exists U \in \tau: x \in U \subseteq N_x \subseteq S$
From [[Set in Discrete Topology is Clopen]]... | Point in Discrete Space is Neighborhood | https://proofwiki.org/wiki/Point_in_Discrete_Space_is_Neighborhood | https://proofwiki.org/wiki/Point_in_Discrete_Space_is_Neighborhood | [
"Discrete Topologies",
"Neighborhoods"
] | [
"Definition:Discrete Topology",
"Definition:Neighborhood (Topology)/Point"
] | [
"Definition:Neighborhood (Topology)/Point",
"Definition:Subset",
"Definition:Open Set/Topology",
"Set in Discrete Topology is Clopen",
"Definition:Open Set/Topology",
"Set is Subset of Itself",
"Definition:Subset",
"Definition:Open Set/Topology"
] |
proofwiki-3605 | Discrete Space is Strongly Locally Compact | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is strongly locally compact. | From Point in Discrete Space is Neighborhood, every point $x \in S$ is contained in an open set $\set x$ of $T$.
Then from Interior Equals Closure of Subset of Discrete Space we have that $\set x$ equals its closure in $T$.
From Singleton Set in Discrete Space is Compact, we have that $\set x$ is compact in $T$.
Hence ... | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Space|discrete topological space]].
Then $T$ is [[Definition:Strongly Locally Compact Space|strongly locally compact]]. | From [[Point in Discrete Space is Neighborhood]], every point $x \in S$ is contained in an [[Definition:Open Set (Topology)|open set]] $\set x$ of $T$.
Then from [[Interior Equals Closure of Subset of Discrete Space]] we have that $\set x$ equals its [[Definition:Closure (Topology)|closure]] in $T$.
From [[Singleton ... | Discrete Space is Strongly Locally Compact | https://proofwiki.org/wiki/Discrete_Space_is_Strongly_Locally_Compact | https://proofwiki.org/wiki/Discrete_Space_is_Strongly_Locally_Compact | [
"Discrete Topologies",
"Examples of Strongly Locally Compact Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Strongly Locally Compact Space"
] | [
"Point in Discrete Space is Neighborhood",
"Definition:Open Set/Topology",
"Interior Equals Closure of Subset of Discrete Space",
"Definition:Closure (Topology)",
"Singleton Set in Discrete Space is Compact",
"Definition:Compact Topological Space/Subspace",
"Definition:Strongly Locally Compact Space"
] |
proofwiki-3606 | Discrete Space is First-Countable | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is first-countable. | From Point in Discrete Space is Neighborhood, every point $x \in S$ is contained in an open set $\set x$.
From the definition of local basis, it is clear that $\set {\set x}$ is (trivially) a local basis at $x$.
That is, that every open set of $S$ containing $x$ also contains at least one of the sets of $\set {\set x}$... | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Topology|discrete topological space]].
Then $T$ is [[Definition:First-Countable Space|first-countable]]. | From [[Point in Discrete Space is Neighborhood]], every point $x \in S$ is contained in an [[Definition:Open Set (Topology)|open set]] $\set x$.
From the definition of [[Definition:Local Basis|local basis]], it is clear that $\set {\set x}$ is (trivially) a [[Definition:Local Basis|local basis]] at $x$.
That is, that... | Discrete Space is First-Countable | https://proofwiki.org/wiki/Discrete_Space_is_First-Countable | https://proofwiki.org/wiki/Discrete_Space_is_First-Countable | [
"Discrete Topologies",
"Examples of First-Countable Spaces"
] | [
"Definition:Discrete Topology",
"Definition:First-Countable Space"
] | [
"Point in Discrete Space is Neighborhood",
"Definition:Open Set/Topology",
"Definition:Local Basis",
"Definition:Local Basis",
"Definition:Open Set/Topology",
"Definition:Set",
"Definition:Countable Set",
"Definition:First-Countable Space"
] |
proofwiki-3607 | Discrete Space has Open Locally Finite Cover | Let $T = \struct {S, \tau}$ be a discrete topological space.
Consider the set $\CC$ of all singleton subsets of $S$:
:$\CC := \set {\set x: x \in S}$
Then $\CC$ is an open cover of $T$ which is locally finite.
This cover is the finest cover on $S$.
That is, if $\VV$ is a cover of $T$, then $\CC$ is a refinement of $\VV... | We have that:
:$\forall x \in S: \exists \set x \in \CC: x \in \set x$
and so $\CC$ is a cover for $S$.
Then from Set in Discrete Topology is Clopen, it follows that $\CC$ is an open cover of $T$.
From Point in Discrete Space is Neighborhood, every point $x \in S$ has a neighborhood $\set x$.
This neighborhood $\set x$... | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Topology|discrete topological space]].
Consider the [[Definition:Set|set]] $\CC$ of all [[Definition:Singleton|singleton]] [[Definition:Subset|subsets]] of $S$:
:$\CC := \set {\set x: x \in S}$
Then $\CC$ is an [[Definition:Open Cover|open cover]] of $T$ which i... | We have that:
:$\forall x \in S: \exists \set x \in \CC: x \in \set x$
and so $\CC$ is a [[Definition:Cover of Set|cover]] for $S$.
Then from [[Set in Discrete Topology is Clopen]], it follows that $\CC$ is an [[Definition:Open Cover|open cover]] of $T$.
From [[Point in Discrete Space is Neighborhood]], every point $... | Discrete Space has Open Locally Finite Cover | https://proofwiki.org/wiki/Discrete_Space_has_Open_Locally_Finite_Cover | https://proofwiki.org/wiki/Discrete_Space_has_Open_Locally_Finite_Cover | [
"Discrete Topologies",
"Covers"
] | [
"Definition:Discrete Topology",
"Definition:Set",
"Definition:Singleton",
"Definition:Subset",
"Definition:Open Cover",
"Definition:Locally Finite Cover",
"Definition:Cover of Set",
"Definition:Refinement of Cover/Finer Cover",
"Definition:Cover of Set",
"Definition:Refinement of Cover"
] | [
"Definition:Cover of Set",
"Set in Discrete Topology is Clopen",
"Definition:Open Cover",
"Point in Discrete Space is Neighborhood",
"Definition:Neighborhood (Topology)/Point",
"Definition:Neighborhood (Topology)/Point",
"Definition:Set Intersection",
"Definition:Element",
"Definition:Finite Set",
... |
proofwiki-3608 | Discrete Space is Paracompact | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is paracompact. | Let $\VV$ be any open cover of $S$.
Consider the set $\CC$ of all singleton subsets of $S$:
:$\CC := \set {\set x: x \in S}$
From Discrete Space has Open Locally Finite Cover, $\CC$ is an open cover which is locally finite.
This result also shows that $\CC$ is the finest cover on $T$.
So $\CC$ is an open refinement of ... | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Topology|discrete topological space]].
Then $T$ is [[Definition:Paracompact Space|paracompact]]. | Let $\VV$ be any [[Definition:Open Cover|open cover]] of $S$.
Consider the [[Definition:Set|set]] $\CC$ of all [[Definition:Singleton|singleton]] [[Definition:Subset|subsets]] of $S$:
:$\CC := \set {\set x: x \in S}$
From [[Discrete Space has Open Locally Finite Cover]], $\CC$ is an [[Definition:Open Cover|open cove... | Discrete Space is Paracompact | https://proofwiki.org/wiki/Discrete_Space_is_Paracompact | https://proofwiki.org/wiki/Discrete_Space_is_Paracompact | [
"Discrete Topologies",
"Examples of Paracompact Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Paracompact Space"
] | [
"Definition:Open Cover",
"Definition:Set",
"Definition:Singleton",
"Definition:Subset",
"Discrete Space has Open Locally Finite Cover",
"Definition:Open Cover",
"Definition:Locally Finite Cover",
"Definition:Refinement of Cover/Finer Cover",
"Definition:Open Refinement",
"Definition:Locally Finite... |
proofwiki-3609 | Singleton Set in Discrete Space is Compact | Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$.
Let $x \in S$.
Then $\set x$ is compact. | From Point in Discrete Space is Neighborhood, every point $x \in S$ is contained in an open set $\set x$.
Then from Interior Equals Closure of Subset of Discrete Space we have that $\set x$ equals its closure.
As $\set {\set x}$ is (trivially) an open cover of $\set x$, it follows by definition that $\set x$ is compact... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] where $\tau$ is the [[Definition:Discrete Topology|discrete topology]] on $S$.
Let $x \in S$.
Then $\set x$ is [[Definition:Compact Topological Subspace|compact]]. | From [[Point in Discrete Space is Neighborhood]], every point $x \in S$ is contained in an [[Definition:Open Set (Topology)|open set]] $\set x$.
Then from [[Interior Equals Closure of Subset of Discrete Space]] we have that $\set x$ equals its [[Definition:Closure (Topology)|closure]].
As $\set {\set x}$ is (triviall... | Singleton Set in Discrete Space is Compact/Proof 1 | https://proofwiki.org/wiki/Singleton_Set_in_Discrete_Space_is_Compact | https://proofwiki.org/wiki/Singleton_Set_in_Discrete_Space_is_Compact/Proof_1 | [
"Singleton Set in Discrete Space is Compact",
"Discrete Topologies",
"Compact Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Discrete Topology",
"Definition:Compact Topological Space/Subspace"
] | [
"Point in Discrete Space is Neighborhood",
"Definition:Open Set/Topology",
"Interior Equals Closure of Subset of Discrete Space",
"Definition:Closure (Topology)",
"Definition:Open Cover",
"Definition:Compact Topological Space"
] |
proofwiki-3610 | Singleton Set in Discrete Space is Compact | Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$.
Let $x \in S$.
Then $\set x$ is compact. | Follows directly from Finite Topological Space is Compact.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] where $\tau$ is the [[Definition:Discrete Topology|discrete topology]] on $S$.
Let $x \in S$.
Then $\set x$ is [[Definition:Compact Topological Subspace|compact]]. | Follows directly from [[Finite Topological Space is Compact]].
{{qed}} | Singleton Set in Discrete Space is Compact/Proof 2 | https://proofwiki.org/wiki/Singleton_Set_in_Discrete_Space_is_Compact | https://proofwiki.org/wiki/Singleton_Set_in_Discrete_Space_is_Compact/Proof_2 | [
"Singleton Set in Discrete Space is Compact",
"Discrete Topologies",
"Compact Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Discrete Topology",
"Definition:Compact Topological Space/Subspace"
] | [
"Finite Topological Space is Compact"
] |
proofwiki-3611 | Countable Discrete Space is Sigma-Compact | Let $T = \struct {S, \tau}$ be a countable discrete topological space.
Then $T$ is $\sigma$-compact. | We have that Singleton Set in Discrete Space is Compact.
We also have that $S$ is the union of all its singleton sets:
:$\ds S = \bigcup_{x \mathop \in S} \set x$
As $S$ is countable, it is the union of countably many compact sets.
Hence the result, by definition of $\sigma$-compact.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Countable Discrete Space|countable discrete topological space]].
Then $T$ is [[Definition:Sigma-Compact Space|$\sigma$-compact]]. | We have that [[Singleton Set in Discrete Space is Compact]].
We also have that $S$ is the [[Definition:Set Union|union]] of all its [[Definition:Singleton|singleton sets]]:
:$\ds S = \bigcup_{x \mathop \in S} \set x$
As $S$ is [[Definition:Countable Set|countable]], it is the [[Definition:Set Union|union]] of [[Defin... | Countable Discrete Space is Sigma-Compact/Proof 1 | https://proofwiki.org/wiki/Countable_Discrete_Space_is_Sigma-Compact | https://proofwiki.org/wiki/Countable_Discrete_Space_is_Sigma-Compact/Proof_1 | [
"Countable Discrete Space is Sigma-Compact",
"Countable Discrete Topologies",
"Examples of Sigma-Compact Spaces"
] | [
"Definition:Discrete Topology/Countable",
"Definition:Sigma-Compact Space"
] | [
"Singleton Set in Discrete Space is Compact",
"Definition:Set Union",
"Definition:Singleton",
"Definition:Countable Set",
"Definition:Set Union",
"Definition:Countable Set",
"Definition:Compact Topological Space/Subspace",
"Definition:Sigma-Compact Space"
] |
proofwiki-3612 | Countable Discrete Space is Sigma-Compact | Let $T = \struct {S, \tau}$ be a countable discrete topological space.
Then $T$ is $\sigma$-compact. | A direct application of Countable Space is Sigma-Compact. | Let $T = \struct {S, \tau}$ be a [[Definition:Countable Discrete Space|countable discrete topological space]].
Then $T$ is [[Definition:Sigma-Compact Space|$\sigma$-compact]]. | A direct application of [[Countable Space is Sigma-Compact]]. | Countable Discrete Space is Sigma-Compact/Proof 2 | https://proofwiki.org/wiki/Countable_Discrete_Space_is_Sigma-Compact | https://proofwiki.org/wiki/Countable_Discrete_Space_is_Sigma-Compact/Proof_2 | [
"Countable Discrete Space is Sigma-Compact",
"Countable Discrete Topologies",
"Examples of Sigma-Compact Spaces"
] | [
"Definition:Discrete Topology/Countable",
"Definition:Sigma-Compact Space"
] | [
"Countable Space is Sigma-Compact"
] |
proofwiki-3613 | Countable Discrete Space is Lindelöf | Let $T = \struct {S, \tau}$ be a countable discrete topological space.
Then $T$ is a Lindelöf space. | We have:
:Countable Discrete Space is $\sigma$-Compact
:$\sigma$-Compact Space is Lindelöf Space
So if $S$ is countable, $T$ is a Lindelöf space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Countable Discrete Topology|countable discrete topological space]].
Then $T$ is a [[Definition:Lindelöf Space|Lindelöf space]]. | We have:
:[[Countable Discrete Space is Sigma-Compact|Countable Discrete Space is $\sigma$-Compact]]
:[[Sigma-Compact Space is Lindelöf Space|$\sigma$-Compact Space is Lindelöf Space]]
So if $S$ is [[Definition:Countable Set|countable]], $T$ is a [[Definition:Lindelöf Space|Lindelöf space]].
{{qed}} | Countable Discrete Space is Lindelöf | https://proofwiki.org/wiki/Countable_Discrete_Space_is_Lindelöf | https://proofwiki.org/wiki/Countable_Discrete_Space_is_Lindelöf | [
"Countable Discrete Topologies",
"Examples of Lindelöf Spaces"
] | [
"Definition:Discrete Topology/Countable",
"Definition:Lindelöf Space"
] | [
"Countable Discrete Space is Sigma-Compact",
"Sigma-Compact Space is Lindelöf",
"Definition:Countable Set",
"Definition:Lindelöf Space"
] |
proofwiki-3614 | Basis for Discrete Topology | Let $S$ be a set.
Let $\tau$ be the discrete topology on $S$.
Let $\BB$ be the set of all singleton subsets of $S$:
:$\BB := \set {\set x: x \in S}$.
Then $\BB$ is a basis for $T$. | Let $T = \struct {S, \tau}$ be the discrete space on $S$.
Let $U \in \tau$.
Then:
:$\ds U = \bigcup_{x \mathop \in U} \set x$
Hence:
:$\forall x \in U: \exists \set x \in \BB: \set x \subseteq U$
Thus $U$ is the union of elements of $\BB$.
Hence by definition $\BB$ is a basis for $T$.
{{qed}}
Category:Discrete Topolog... | Let $S$ be a [[Definition:Set|set]].
Let $\tau$ be the [[Definition:Discrete Topology|discrete topology]] on $S$.
Let $\BB$ be the set of all [[Definition:Singleton|singleton]] subsets of $S$:
:$\BB := \set {\set x: x \in S}$.
Then $\BB$ is a [[Definition:Basis (Topology)|basis]] for $T$. | Let $T = \struct {S, \tau}$ be the [[Definition:Discrete Space|discrete space]] on $S$.
Let $U \in \tau$.
Then:
:$\ds U = \bigcup_{x \mathop \in U} \set x$
Hence:
:$\forall x \in U: \exists \set x \in \BB: \set x \subseteq U$
Thus $U$ is the [[Definition:Set Union|union]] of [[Definition:Element|elements]] of $\B... | Basis for Discrete Topology | https://proofwiki.org/wiki/Basis_for_Discrete_Topology | https://proofwiki.org/wiki/Basis_for_Discrete_Topology | [
"Discrete Topologies",
"Examples of Topological Bases"
] | [
"Definition:Set",
"Definition:Discrete Topology",
"Definition:Singleton",
"Definition:Basis (Topology)"
] | [
"Definition:Discrete Topology",
"Definition:Set Union",
"Definition:Element",
"Definition:Basis (Topology)",
"Category:Discrete Topologies",
"Category:Examples of Topological Bases"
] |
proofwiki-3615 | Countable Discrete Space is Second-Countable | Let $T = \struct {S, \tau}$ be a countable discrete topological space.
Then $T$ is second-countable. | From Basis for Discrete Topology, the set:
:$\BB := \set {\set x: x \in S}$
is a basis for $T$.
There is a trivial one-to-one correspondence $\phi: S \leftrightarrow \BB$ between $S$ and $\BB$:
:$\forall x \in S: \map \phi x = \set x$
Let $S$ be countable.
Then $\BB$ is also countable by definition of countability.
So ... | Let $T = \struct {S, \tau}$ be a [[Definition:Countable Discrete Topology|countable discrete topological space]].
Then $T$ is [[Definition:Second-Countable Space|second-countable]]. | From [[Basis for Discrete Topology]], the set:
:$\BB := \set {\set x: x \in S}$
is a [[Definition:Basis (Topology)|basis]] for $T$.
There is a trivial [[Definition:Bijection|one-to-one correspondence]] $\phi: S \leftrightarrow \BB$ between $S$ and $\BB$:
:$\forall x \in S: \map \phi x = \set x$
Let $S$ be [[Definiti... | Countable Discrete Space is Second-Countable | https://proofwiki.org/wiki/Countable_Discrete_Space_is_Second-Countable | https://proofwiki.org/wiki/Countable_Discrete_Space_is_Second-Countable | [
"Countable Discrete Topologies",
"Examples of Second-Countable Spaces"
] | [
"Definition:Discrete Topology/Countable",
"Definition:Second-Countable Space"
] | [
"Basis for Discrete Topology",
"Definition:Basis (Topology)",
"Definition:Bijection",
"Definition:Countable Set",
"Definition:Countable Set",
"Definition:Countable Set",
"Definition:Countable Basis",
"Definition:Second-Countable Space"
] |
proofwiki-3616 | Countable Discrete Space is Separable | Let $T = \struct {S, \tau}$ be a countable discrete topological space.
Then $T$ is separable. | Let $T = \left({S, \tau}\right)$ be a countable discrete topological space.
From Countable Discrete Space is Second-Countable:
:$T$ is second-countable.
From Second-Countable Space is Separable:
:$T$ is separable.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Countable Discrete Topology|countable discrete topological space]].
Then $T$ is [[Definition:Separable Space|separable]]. | Let $T = \left({S, \tau}\right)$ be a [[Definition:Countable Discrete Topology|countable discrete topological space]].
From [[Countable Discrete Space is Second-Countable]]:
:$T$ is [[Definition:Second-Countable Space|second-countable]].
From [[Second-Countable Space is Separable]]:
:$T$ is [[Definition:Separable Spa... | Countable Discrete Space is Separable/Proof 1 | https://proofwiki.org/wiki/Countable_Discrete_Space_is_Separable | https://proofwiki.org/wiki/Countable_Discrete_Space_is_Separable/Proof_1 | [
"Countable Discrete Space is Separable",
"Countable Discrete Topologies",
"Examples of Separable Spaces"
] | [
"Definition:Discrete Topology/Countable",
"Definition:Separable Space"
] | [
"Definition:Discrete Topology/Countable",
"Countable Discrete Space is Second-Countable",
"Definition:Second-Countable Space",
"Second-Countable Space is Separable",
"Definition:Separable Space"
] |
proofwiki-3617 | Countable Discrete Space is Separable | Let $T = \struct {S, \tau}$ be a countable discrete topological space.
Then $T$ is separable. | Follows immediately from Countable Space is Separable.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Countable Discrete Topology|countable discrete topological space]].
Then $T$ is [[Definition:Separable Space|separable]]. | Follows immediately from [[Countable Space is Separable]].
{{qed}} | Countable Discrete Space is Separable/Proof 2 | https://proofwiki.org/wiki/Countable_Discrete_Space_is_Separable | https://proofwiki.org/wiki/Countable_Discrete_Space_is_Separable/Proof_2 | [
"Countable Discrete Space is Separable",
"Countable Discrete Topologies",
"Examples of Separable Spaces"
] | [
"Definition:Discrete Topology/Countable",
"Definition:Separable Space"
] | [
"Countable Space is Separable"
] |
proofwiki-3618 | Uncountable Discrete Space is not Separable | Let $T = \struct {S, \tau}$ be an uncountable discrete topological space.
Then $T$ is not separable. | {{Recall|Separable Space|separable space}}
{{:Definition:Separable Space}}
Let $H \subseteq S$ be everywhere dense in $T$.
Then by definition of everywhere dense:
:$H^- = S$
where $H^-$ denotes the closure of $H$.
We have {{hypothesis}} that $T$ is a discrete space.
Hence from Interior Equals Closure of Subset of Discr... | Let $T = \struct {S, \tau}$ be an [[Definition:Uncountable Discrete Topology|uncountable discrete topological space]].
Then $T$ is not [[Definition:Separable Space|separable]]. | {{Recall|Separable Space|separable space}}
{{:Definition:Separable Space}}
Let $H \subseteq S$ be [[Definition:Everywhere Dense|everywhere dense]] in $T$.
Then by definition of [[Definition:Everywhere Dense|everywhere dense]]:
:$H^- = S$
where $H^-$ denotes the [[Definition:Closure (Topology)|closure]] of $H$.
We h... | Uncountable Discrete Space is not Separable | https://proofwiki.org/wiki/Uncountable_Discrete_Space_is_not_Separable | https://proofwiki.org/wiki/Uncountable_Discrete_Space_is_not_Separable | [
"Uncountable Discrete Topologies",
"Examples of Separable Spaces",
"Uncountable Sets"
] | [
"Definition:Discrete Topology/Uncountable",
"Definition:Separable Space"
] | [
"Definition:Everywhere Dense",
"Definition:Everywhere Dense",
"Definition:Closure (Topology)",
"Definition:Discrete Topology",
"Interior Equals Closure of Subset of Discrete Space",
"Definition:Uncountable/Set",
"Definition:Countable Set",
"Definition:Everywhere Dense",
"Definition:Separable Space",... |
proofwiki-3619 | Uncountable Discrete Space is not Second-Countable | Let $T = \struct {S, \tau}$ be an uncountable discrete topological space.
Then $T$ is not second-countable. | We have that an Uncountable Discrete Space is not Separable.
From Second-Countable Space is Separable, it follows that $T$ cannot be second-countable.
{{qed}} | Let $T = \struct {S, \tau}$ be an [[Definition:Uncountable Discrete Topology|uncountable discrete topological space]].
Then $T$ is not [[Definition:Second-Countable Space|second-countable]]. | We have that an [[Uncountable Discrete Space is not Separable]].
From [[Second-Countable Space is Separable]], it follows that $T$ cannot be [[Definition:Second-Countable Space|second-countable]].
{{qed}} | Uncountable Discrete Space is not Second-Countable | https://proofwiki.org/wiki/Uncountable_Discrete_Space_is_not_Second-Countable | https://proofwiki.org/wiki/Uncountable_Discrete_Space_is_not_Second-Countable | [
"Uncountable Discrete Topologies",
"Examples of Second-Countable Spaces",
"Uncountable Sets"
] | [
"Definition:Discrete Topology/Uncountable",
"Definition:Second-Countable Space"
] | [
"Uncountable Discrete Space is not Separable",
"Second-Countable Space is Separable",
"Definition:Second-Countable Space"
] |
proofwiki-3620 | Uncountable Discrete Space is not Lindelöf | Let $T = \struct {S, \tau}$ be an uncountable discrete topological space.
Then $T$ is not a Lindelöf space. | Consider the set $\CC$ of all singleton subsets of $S$:
:$\CC := \set {\set x: x \in S}$
From Discrete Space has Open Locally Finite Cover, $\CC$ is an open cover of $S$ which is finer than any other open cover of $S$.
That is, $\CC$ is an open cover of $S$ which is uncountable and has no countable subcover.
(Note that... | Let $T = \struct {S, \tau}$ be an [[Definition:Uncountable Discrete Topology|uncountable discrete topological space]].
Then $T$ is not a [[Definition:Lindelöf Space|Lindelöf space]]. | Consider the [[Definition:Set|set]] $\CC$ of all [[Definition:Singleton|singleton]] [[Definition:Subset|subsets]] of $S$:
:$\CC := \set {\set x: x \in S}$
From [[Discrete Space has Open Locally Finite Cover]], $\CC$ is an [[Definition:Open Cover|open cover]] of $S$ which is [[Definition:Finer Cover|finer]] than any ot... | Uncountable Discrete Space is not Lindelöf | https://proofwiki.org/wiki/Uncountable_Discrete_Space_is_not_Lindelöf | https://proofwiki.org/wiki/Uncountable_Discrete_Space_is_not_Lindelöf | [
"Uncountable Discrete Topologies",
"Examples of Lindelöf Spaces",
"Uncountable Sets"
] | [
"Definition:Discrete Topology/Uncountable",
"Definition:Lindelöf Space"
] | [
"Definition:Set",
"Definition:Singleton",
"Definition:Subset",
"Discrete Space has Open Locally Finite Cover",
"Definition:Open Cover",
"Definition:Refinement of Cover/Finer Cover",
"Definition:Open Cover",
"Definition:Open Cover",
"Definition:Uncountable/Set",
"Definition:Subcover/Countable",
"... |
proofwiki-3621 | Uncountable Discrete Space is not Sigma-Compact | Let $T = \struct {S, \tau}$ be an uncountable discrete topological space.
Then $T$ is not a $\sigma$-compact space. | We have that an Uncountable Discrete Space is not Lindelöf.
But a $\sigma$-compact space is a Lindelöf space.
So an uncountable discrete space cannot be $\sigma$-compact.
{{qed}} | Let $T = \struct {S, \tau}$ be an [[Definition:Uncountable Discrete Space|uncountable discrete topological space]].
Then $T$ is not a [[Definition:Sigma-Compact Space|$\sigma$-compact space]]. | We have that an [[Uncountable Discrete Space is not Lindelöf]].
But a [[Sigma-Compact Space is Lindelöf Space|$\sigma$-compact space is a Lindelöf space]].
So an [[Definition:Uncountable Discrete Space|uncountable discrete space]] cannot be [[Definition:Sigma-Compact Space|$\sigma$-compact]].
{{qed}} | Uncountable Discrete Space is not Sigma-Compact | https://proofwiki.org/wiki/Uncountable_Discrete_Space_is_not_Sigma-Compact | https://proofwiki.org/wiki/Uncountable_Discrete_Space_is_not_Sigma-Compact | [
"Uncountable Discrete Topologies",
"Examples of Sigma-Compact Spaces",
"Uncountable Sets"
] | [
"Definition:Discrete Topology/Uncountable",
"Definition:Sigma-Compact Space"
] | [
"Uncountable Discrete Space is not Lindelöf",
"Sigma-Compact Space is Lindelöf",
"Definition:Discrete Topology/Uncountable",
"Definition:Sigma-Compact Space"
] |
proofwiki-3622 | Discrete Space is Compact iff Finite | Let $T = \struct {S, \tau}$ be a topological space where $\tau$ is the discrete topology on $S$.
Then $T$ is compact {{iff}} $S$ is a finite set, thereby making $\tau$ the finite discrete topology on $S$. | === Necessary Condition ===
Let $T$ be a compact discrete space.
{{AimForCont}} $T$ is infinite.
Let $\CC$ be the cover for $S$ defined as:
:$\CC = \set {\set x: x \in S}$
As $S$ is an infinite set then so is $\CC$.
Let $\CC'$ be a proper subset of $\CC$.
Then:
:$\exists y \in S: \set y \notin \CC'$
and so $\CC'$ is no... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] where $\tau$ is the [[Definition:Discrete Topology|discrete topology]] on $S$.
Then $T$ is [[Definition:Compact Topological Space|compact]] {{iff}} $S$ is a [[Definition:Finite Set|finite set]], thereby making $\tau$ the [[Definition:... | === Necessary Condition ===
Let $T$ be a [[Definition:Compact Topological Space|compact]] [[Definition:Discrete Topology|discrete space]].
{{AimForCont}} $T$ is [[Definition:Infinite Discrete Topology|infinite]].
Let $\CC$ be the [[Definition:Cover of Set|cover]] for $S$ defined as:
:$\CC = \set {\set x: x \in S}$
... | Discrete Space is Compact iff Finite | https://proofwiki.org/wiki/Discrete_Space_is_Compact_iff_Finite | https://proofwiki.org/wiki/Discrete_Space_is_Compact_iff_Finite | [
"Finite Topological Spaces",
"Compact Topological Spaces",
"Discrete Topologies"
] | [
"Definition:Topological Space",
"Definition:Discrete Topology",
"Definition:Compact Topological Space",
"Definition:Finite Set",
"Definition:Discrete Topology/Finite"
] | [
"Definition:Compact Topological Space",
"Definition:Discrete Topology",
"Definition:Discrete Topology/Infinite",
"Definition:Cover of Set",
"Definition:Infinite Set",
"Definition:Proper Subset",
"Definition:Cover of Set",
"Definition:Subcover",
"Definition:Subcover/Finite",
"Definition:Compact Top... |
proofwiki-3623 | Finite Space is Sequentially Compact | Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set.
Then $T$ is sequentially compact. | We have:
:Finite Topological Space is Compact
:Compact Space is Countably Compact
:Finite Space is Second-Countable
:Second-Countable Space is First-Countable
:First-Countable Space is Sequentially Compact iff Countably Compact.
Hence the result.
{{qed}}
Category:Sequentially Compact Spaces
Category:Finite Topological ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] where $S$ is a [[Definition:Finite Set|finite set]].
Then $T$ is [[Definition:Sequentially Compact Space|sequentially compact]]. | We have:
:[[Finite Topological Space is Compact]]
:[[Compact Space is Countably Compact]]
:[[Finite Space is Second-Countable]]
:[[Second-Countable Space is First-Countable]]
:[[First-Countable Space is Sequentially Compact iff Countably Compact]].
Hence the result.
{{qed}}
[[Category:Sequentially Compact Spaces]]
[... | Finite Space is Sequentially Compact | https://proofwiki.org/wiki/Finite_Space_is_Sequentially_Compact | https://proofwiki.org/wiki/Finite_Space_is_Sequentially_Compact | [
"Sequentially Compact Spaces",
"Finite Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Finite Set",
"Definition:Sequentially Compact Space"
] | [
"Finite Topological Space is Compact",
"Compact Space is Countably Compact",
"Finite Space is Second-Countable",
"Second-Countable Space is First-Countable",
"First-Countable Space is Sequentially Compact iff Countably Compact",
"Category:Sequentially Compact Spaces",
"Category:Finite Topological Spaces... |
proofwiki-3624 | Discrete Space is Fully Normal | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is fully normal. | We have that a Discrete Space is fully $T_4$.
Then we note that from Discrete Space satisfies all Separation Properties, a discrete space is a $T_1$ space.
Therefore, by definition, $T$ is fully normal.
{{qed}}
Category:Discrete Topologies
Category:Examples of Fully Normal Spaces
g87snaed81aaupu98m4ycsnw3udifqp | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Topology|discrete topological space]].
Then $T$ is [[Definition:Fully Normal Space|fully normal]]. | We have that a [[Discrete Space is Fully T4|Discrete Space is fully $T_4$]].
Then we note that from [[Discrete Space satisfies all Separation Properties]], a [[Definition:Discrete Space|discrete space]] is a [[Definition:T1 Space|$T_1$ space]].
Therefore, by definition, $T$ is [[Definition:Fully Normal Space|fully no... | Discrete Space is Fully Normal | https://proofwiki.org/wiki/Discrete_Space_is_Fully_Normal | https://proofwiki.org/wiki/Discrete_Space_is_Fully_Normal | [
"Discrete Topologies",
"Examples of Fully Normal Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Fully Normal Space"
] | [
"Discrete Space is Fully T4",
"Discrete Space satisfies all Separation Properties",
"Definition:Discrete Topology",
"Definition:T1 Space",
"Definition:Fully Normal Space",
"Category:Discrete Topologies",
"Category:Examples of Fully Normal Spaces"
] |
proofwiki-3625 | Discrete Space is Fully T4 | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is fully $T_4$. | Consider the set $\CC$ of all singleton subsets of $S$:
:$\CC := \set {\set x: x \in S}$
From Discrete Space has Open Locally Finite Cover:
:$\CC$ is an open cover of $T$ which is locally finite.
Let $x \in S$.
The star of $x$ with respect to $\CC$ is defined as:
:$\ds x^* := \bigcup \set {U \in \CC: x \in U}$
That is,... | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Topology|discrete topological space]].
Then $T$ is [[Definition:Fully T4 Space|fully $T_4$]]. | Consider the [[Definition:Set|set]] $\CC$ of all [[Definition:Singleton|singleton]] [[Definition:Subset|subsets]] of $S$:
:$\CC := \set {\set x: x \in S}$
From [[Discrete Space has Open Locally Finite Cover]]:
:$\CC$ is an [[Definition:Open Cover|open cover]] of $T$ which is [[Definition:Locally Finite Cover|locally f... | Discrete Space is Fully T4 | https://proofwiki.org/wiki/Discrete_Space_is_Fully_T4 | https://proofwiki.org/wiki/Discrete_Space_is_Fully_T4 | [
"Discrete Topologies",
"Examples of Fully T4 Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Fully T4 Space"
] | [
"Definition:Set",
"Definition:Singleton",
"Definition:Subset",
"Discrete Space has Open Locally Finite Cover",
"Definition:Open Cover",
"Definition:Locally Finite Cover",
"Definition:Star (Topology)",
"Definition:Set Union",
"Definition:Open Cover",
"Definition:Cover of Set",
"Definition:Star (T... |
proofwiki-3626 | Finite Topological Space is Compact | Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set.
Then $T$ is compact. | From Cardinality of Empty Set, the empty set is finite.
Suppose $S$ is empty.
Then by Empty Set is Compact, $T$ is compact.
Suppose $S$ is non-empty.
Let $\VV$ be an open cover of $T$.
For each $x \in S$, define $\VV_x$ to be $\set {V \in \VV : x \in V}$
Since $S$ is finite, and since by definition of a cover, each $x\... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] where $S$ is a [[Definition:Finite Set|finite set]].
Then $T$ is [[Definition:Compact Topological Space|compact]]. | From [[Cardinality of Empty Set]], the [[Definition:Empty Set|empty set]] is [[Definition:Finite Set|finite]].
Suppose $S$ is [[Definition:Empty Set|empty]].
Then by [[Empty Set is Compact]], $T$ is [[Definition:Compact Topological Space|compact]].
Suppose $S$ is [[Definition:Non-Empty Set|non-empty]].
Let $\VV$ be... | Finite Topological Space is Compact/Proof 1 | https://proofwiki.org/wiki/Finite_Topological_Space_is_Compact | https://proofwiki.org/wiki/Finite_Topological_Space_is_Compact/Proof_1 | [
"Finite Topological Space is Compact",
"Finite Topological Spaces",
"Compact Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Finite Set",
"Definition:Compact Topological Space"
] | [
"Cardinality of Empty Set",
"Definition:Empty Set",
"Definition:Finite Set",
"Definition:Empty Set",
"Empty Set is Compact",
"Definition:Compact Topological Space",
"Definition:Non-Empty Set",
"Definition:Open Cover",
"Definition:Finite Set",
"Definition:Finite Set",
"Principle of Finite Choice"... |
proofwiki-3627 | Finite Topological Space is Compact | Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set.
Then $T$ is compact. | From Power Set of Finite Set is Finite, the power set of $S$ is finite.
Since the topology $\tau$ is by definition a set of subsets of $S$, it follows that $\tau$ is finite as well.
Let $\VV$ be an open cover of $S$.
By definition $\VV \subseteq \tau$ and so is also a finite set.
From Set is Subset of Itself, $\VV \sub... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] where $S$ is a [[Definition:Finite Set|finite set]].
Then $T$ is [[Definition:Compact Topological Space|compact]]. | From [[Power Set of Finite Set is Finite]], the [[Definition:Power Set|power set]] of $S$ is [[Definition:Finite Set|finite]].
Since the [[Definition:Topology|topology]] $\tau$ is by definition a [[Definition:Set of Sets|set]] of [[Definition:Subset|subsets]] of $S$, it follows that $\tau$ is [[Definition:Finite Set|f... | Finite Topological Space is Compact/Proof 2 | https://proofwiki.org/wiki/Finite_Topological_Space_is_Compact | https://proofwiki.org/wiki/Finite_Topological_Space_is_Compact/Proof_2 | [
"Finite Topological Space is Compact",
"Finite Topological Spaces",
"Compact Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Finite Set",
"Definition:Compact Topological Space"
] | [
"Power Set of Finite Set is Finite",
"Definition:Power Set",
"Definition:Finite Set",
"Definition:Topology",
"Definition:Set of Sets",
"Definition:Subset",
"Definition:Finite Set",
"Definition:Open Cover",
"Definition:Finite Set",
"Set is Subset of Itself",
"Definition:Subcover/Finite",
"Defin... |
proofwiki-3628 | Finite Space is Second-Countable | Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set.
Then $T$ is a second-countable space. | Let $T = \struct {S, \tau}$ be a topological space where $S$ is finite.
As $S$ is finite, then so is its power set $\powerset S$.
So is $\tau \subseteq \powerset S$.
Now $\tau$ is a basis for itself, as every set in $\tau$ is a union of sets from $\tau$ (just one).
But $\tau$ is finite, and so countable.
The result fol... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] where $S$ is a [[Definition:Finite Set|finite set]].
Then $T$ is a [[Definition:Second-Countable Space|second-countable space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] where $S$ is [[Definition:Finite Set|finite]].
As $S$ is [[Definition:Finite Set|finite]], then so is its [[Definition:Power Set|power set]] $\powerset S$.
So is $\tau \subseteq \powerset S$.
Now $\tau$ is a [[Definition:Basis (Topol... | Finite Space is Second-Countable | https://proofwiki.org/wiki/Finite_Space_is_Second-Countable | https://proofwiki.org/wiki/Finite_Space_is_Second-Countable | [
"Second-Countable Spaces"
] | [
"Definition:Topological Space",
"Definition:Finite Set",
"Definition:Second-Countable Space"
] | [
"Definition:Topological Space",
"Definition:Finite Set",
"Definition:Finite Set",
"Definition:Power Set",
"Definition:Basis (Topology)",
"Definition:Set Union",
"Definition:Finite Set",
"Definition:Countable Set",
"Definition:Second-Countable Space",
"Category:Second-Countable Spaces"
] |
proofwiki-3629 | Finite Discrete Space satisfies all Compactness Properties | Let $T = \struct {S, \tau}$ be a finite discrete topological space.
Then $T$ satisfies the following compactness properties:
:$T$ is compact.
:$T$ is Sequentially Compact.
:$T$ is Countably Compact.
:$T$ is Weakly Countably Compact.
:$T$ is a Lindelöf Space
:$T$ is Pseudocompact.
:$T$ is $\sigma$-Compact.
:$T$ is Local... | A finite discrete space is by definition a topology on a finite set.
A Discrete Space is Fully Normal.
A fully normal space is fully $T_4$ by definition.
The rest of the results follow directly from Finite Space Satisfies All Compactness Properties. | Let $T = \struct {S, \tau}$ be a [[Definition:Finite Discrete Topology|finite discrete topological space]].
Then $T$ satisfies the following [[Definition:Compact Topological Space|compactness properties]]:
:$T$ is [[Definition:Compact Topological Space|compact]].
:$T$ is [[Definition:Sequentially Compact Space|Seque... | A [[Definition:Finite Discrete Topology|finite discrete space]] is by definition a [[Definition:Topology|topology]] on a [[Definition:Finite Set|finite set]].
A [[Discrete Space is Fully Normal]].
A [[Definition:Fully Normal Space|fully normal space]] is [[Definition:Fully T4 Space|fully $T_4$]] by definition.
The ... | Finite Discrete Space satisfies all Compactness Properties | https://proofwiki.org/wiki/Finite_Discrete_Space_satisfies_all_Compactness_Properties | https://proofwiki.org/wiki/Finite_Discrete_Space_satisfies_all_Compactness_Properties | [
"Finite Discrete Topologies",
"Finite Topological Spaces",
"Topological Compactness"
] | [
"Definition:Discrete Topology/Finite",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space",
"Definition:Sequentially Compact Space",
"Definition:Countably Compact Space",
"Definition:Weakly Countably Compact Space",
"Definition:Lindelöf Space",
"Definition:Pseudocompact Spac... | [
"Definition:Discrete Topology/Finite",
"Definition:Topology",
"Definition:Finite Set",
"Discrete Space is Fully Normal",
"Definition:Fully Normal Space",
"Definition:Fully T4 Space",
"Finite Space Satisfies All Compactness Properties"
] |
proofwiki-3630 | Finite Space Satisfies All Compactness Properties | Let $T = \struct {S, \tau}$ be a topological space where $S$ is a finite set.
Then $T$ satisfies the following compactness properties:
:$T$ is compact.
:$T$ is sequentially compact.
:$T$ is countably compact.
:$T$ is weakly countably compact.
:$T$ is a Lindelöf space.
:$T$ is pseudocompact.
:$T$ is $\sigma$-compact.
:$... | We have that:
:A Finite Topological Space is Compact.
:A Finite Space is Sequentially Compact.
The remaining properties are demonstrated in:
:Sequence of Implications of Global Compactness Properties
:Sequence of Implications of Local Compactness Properties
:Sequence of Implications of Paracompactness Properties
{{qed}... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] where $S$ is a [[Definition:Finite Set|finite set]].
Then $T$ satisfies the following [[Definition:Compact Topological Space|compactness properties]]:
:$T$ is [[Definition:Compact Topological Space|compact]].
:$T$ is [[Definition:Sequ... | We have that:
:A [[Finite Topological Space is Compact]].
:A [[Finite Space is Sequentially Compact]].
The remaining properties are demonstrated in:
:[[Sequence of Implications of Global Compactness Properties]]
:[[Sequence of Implications of Local Compactness Properties]]
:[[Sequence of Implications of Paracompact... | Finite Space Satisfies All Compactness Properties | https://proofwiki.org/wiki/Finite_Space_Satisfies_All_Compactness_Properties | https://proofwiki.org/wiki/Finite_Space_Satisfies_All_Compactness_Properties | [
"Finite Topological Spaces",
"Compact Topological Spaces"
] | [
"Definition:Topological Space",
"Definition:Finite Set",
"Definition:Compact Topological Space",
"Definition:Compact Topological Space",
"Definition:Sequentially Compact Space",
"Definition:Countably Compact Space",
"Definition:Weakly Countably Compact Space",
"Definition:Lindelöf Space",
"Definitio... | [
"Finite Topological Space is Compact",
"Finite Space is Sequentially Compact",
"Sequence of Implications of Global Compactness Properties",
"Sequence of Implications of Local Compactness Properties",
"Sequence of Implications of Paracompactness Properties",
"Category:Finite Topological Spaces",
"Categor... |
proofwiki-3631 | Discrete Space is Non-Meager | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is non-meager. | Let $U \subseteq S$ such that $U \ne \O$.
From Interior Equals Closure of Subset of Discrete Space, we have:
:$U^\circ = U = U^-$
where $U^\circ$ is the interior and $U^-$ the closure of $U$.
So:
:$\paren {U^-}^\circ = U \ne \O$
Thus, by definition, no non-empty subset of $S$ is nowhere dense.
So $S$ can not be the uni... | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Topology|discrete topological space]].
Then $T$ is [[Definition:Non-Meager Space|non-meager]]. | Let $U \subseteq S$ such that $U \ne \O$.
From [[Interior Equals Closure of Subset of Discrete Space]], we have:
:$U^\circ = U = U^-$
where $U^\circ$ is the [[Definition:Interior (Topology)|interior]] and $U^-$ the [[Definition:Closure (Topology)|closure]] of $U$.
So:
:$\paren {U^-}^\circ = U \ne \O$
Thus, by defin... | Discrete Space is Non-Meager/Proof 1 | https://proofwiki.org/wiki/Discrete_Space_is_Non-Meager | https://proofwiki.org/wiki/Discrete_Space_is_Non-Meager/Proof_1 | [
"Discrete Space is Non-Meager",
"Discrete Topologies",
"Examples of Non-Meager Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Meager Space/Non-Meager"
] | [
"Interior Equals Closure of Subset of Discrete Space",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Nowhere Dense",
"Definition:Set Union",
"Definition:Countable Set",
"Definition:Nowhere Dense",
"Definition:Subset"... |
proofwiki-3632 | Discrete Space is Non-Meager | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is non-meager. | Let $x \in S$ be an arbitrary point of $T$.
From Set in Discrete Topology is Clopen it follows that $\set x$ is open in $T$.
The result follows from Space with Open Point is Non-Meager.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Topology|discrete topological space]].
Then $T$ is [[Definition:Non-Meager Space|non-meager]]. | Let $x \in S$ be an arbitrary point of $T$.
From [[Set in Discrete Topology is Clopen]] it follows that $\set x$ is [[Definition:Open Set (Topology)|open]] in $T$.
The result follows from [[Space with Open Point is Non-Meager]].
{{qed}} | Discrete Space is Non-Meager/Proof 2 | https://proofwiki.org/wiki/Discrete_Space_is_Non-Meager | https://proofwiki.org/wiki/Discrete_Space_is_Non-Meager/Proof_2 | [
"Discrete Space is Non-Meager",
"Discrete Topologies",
"Examples of Non-Meager Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Meager Space/Non-Meager"
] | [
"Set in Discrete Topology is Clopen",
"Definition:Open Set/Topology",
"Space with Open Point is Non-Meager"
] |
proofwiki-3633 | Discrete Space is Complete Metric Space | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is a complete metric space. | Let $d: S \times S \to \R$ be the standard discrete metric.
From Standard Discrete Metric induces Discrete Topology we have that $\struct {S, d}$ is a metric space whose induced topology is $\tau$.
Consider now a Cauchy sequence $\sequence {x_n}_{n \mathop \in \N}$ in $S$.
By the definition of Cauchy sequence:
:$\foral... | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Topology|discrete topological space]].
Then $T$ is a [[Definition:Complete Metric Space|complete metric space]]. | Let $d: S \times S \to \R$ be the [[Definition:Standard Discrete Metric|standard discrete metric]].
From [[Standard Discrete Metric induces Discrete Topology]] we have that $\struct {S, d}$ is a [[Definition:Metric Space|metric space]] whose [[Definition:Topology Induced by Metric|induced topology]] is $\tau$.
Consid... | Discrete Space is Complete Metric Space | https://proofwiki.org/wiki/Discrete_Space_is_Complete_Metric_Space | https://proofwiki.org/wiki/Discrete_Space_is_Complete_Metric_Space | [
"Discrete Topologies",
"Standard Discrete Metric",
"Examples of Complete Metric Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Complete Metric Space"
] | [
"Definition:Standard Discrete Metric",
"Standard Discrete Metric induces Discrete Topology",
"Definition:Metric Space",
"Definition:Topology Induced by Metric",
"Definition:Cauchy Sequence",
"Definition:Cauchy Sequence",
"Definition:Standard Discrete Metric",
"Definition:Sequence",
"Definition:Event... |
proofwiki-3634 | Discrete Space is not Dense-In-Itself | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is not dense-in-itself. | By definition, $T$ is dense-in-itself {{iff}} it contains no isolated points.
The result follows from Topological Space is Discrete iff All Points are Isolated.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Topology|discrete topological space]].
Then $T$ is not [[Definition:Dense-in-itself|dense-in-itself]]. | By definition, $T$ is [[Definition:Dense-in-itself|dense-in-itself]] {{iff}} it contains no [[Definition:Isolated Point (Topology)|isolated points]].
The result follows from [[Topological Space is Discrete iff All Points are Isolated]].
{{qed}} | Discrete Space is not Dense-In-Itself | https://proofwiki.org/wiki/Discrete_Space_is_not_Dense-In-Itself | https://proofwiki.org/wiki/Discrete_Space_is_not_Dense-In-Itself | [
"Discrete Topologies",
"Examples of Dense-in-itself"
] | [
"Definition:Discrete Topology",
"Definition:Dense-in-itself"
] | [
"Definition:Dense-in-itself",
"Definition:Isolated Point (Topology)",
"Topological Space is Discrete iff All Points are Isolated"
] |
proofwiki-3635 | Non-Trivial Discrete Space is not Connected | Let $T = \struct {S, \tau}$ be a non-trivial discrete topological space.
Then $T$ is not connected. | Let $T = \struct {S, \tau}$ be a non-trivial discrete space.
Let $a \in S$.
Let $A = \set a$ and $B = \relcomp S {\set a}$, where $\relcomp S {\set a}$ is the complement of $A$ in $S$.
As $T$ is not trivial, $B \ne \O$.
Then from Set in Discrete Topology is Clopen, $A$ and $B$ are both open.
So $A \mid B$ is a separati... | Let $T = \struct {S, \tau}$ be a non-[[Definition:Trivial Topological Space|trivial]] [[Definition:Discrete Topology|discrete topological space]].
Then $T$ is not [[Definition:Connected Topological Space|connected]]. | Let $T = \struct {S, \tau}$ be a non-[[Definition:Trivial Topological Space|trivial]] [[Definition:Discrete Space|discrete space]].
Let $a \in S$.
Let $A = \set a$ and $B = \relcomp S {\set a}$, where $\relcomp S {\set a}$ is the [[Definition:Relative Complement|complement]] of $A$ in $S$.
As $T$ is not [[Definition... | Non-Trivial Discrete Space is not Connected | https://proofwiki.org/wiki/Non-Trivial_Discrete_Space_is_not_Connected | https://proofwiki.org/wiki/Non-Trivial_Discrete_Space_is_not_Connected | [
"Non-Trivial Discrete Space is not Connected",
"Discrete Topologies",
"Examples of Connected Topological Spaces"
] | [
"Definition:Trivial Topological Space",
"Definition:Discrete Topology",
"Definition:Connected Topological Space"
] | [
"Definition:Trivial Topological Space",
"Definition:Discrete Topology",
"Definition:Relative Complement",
"Definition:Trivial Topological Space",
"Set in Discrete Topology is Clopen",
"Definition:Open Set/Topology",
"Definition:Separation (Topology)",
"Definition:Connected Topological Space"
] |
proofwiki-3636 | Discrete Space is Locally Path-Connected | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is locally path-connected. | From Set in Discrete Topology is Clopen, $\set a$ is open in $T$.
From Basis for Discrete Topology, the set:
:$\BB := \set {\set x: x \in S}$
is a basis for $T$.
Let $\set x \in \BB$.
From Point is Path-Connected to Itself, it follows that $\set x$ is path-connected.
Hence $T$ has a basis consisting entirely of path-co... | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Topology|discrete topological space]].
Then $T$ is [[Definition:Locally Path-Connected Space|locally path-connected]]. | From [[Set in Discrete Topology is Clopen]], $\set a$ is [[Definition:Open Set (Topology)|open]] in $T$.
From [[Basis for Discrete Topology]], the set:
:$\BB := \set {\set x: x \in S}$
is a [[Definition:Basis (Topology)|basis]] for $T$.
Let $\set x \in \BB$.
From [[Point is Path-Connected to Itself]], it follows th... | Discrete Space is Locally Path-Connected | https://proofwiki.org/wiki/Discrete_Space_is_Locally_Path-Connected | https://proofwiki.org/wiki/Discrete_Space_is_Locally_Path-Connected | [
"Discrete Topologies",
"Examples of Locally Path-Connected Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Locally Path-Connected Space"
] | [
"Set in Discrete Topology is Clopen",
"Definition:Open Set/Topology",
"Basis for Discrete Topology",
"Definition:Basis (Topology)",
"Point is Path-Connected to Itself",
"Definition:Path-Connected/Set",
"Definition:Basis (Topology)",
"Definition:Path-Connected/Set",
"Definition:Locally Path-Connected... |
proofwiki-3637 | Point is Path-Connected to Itself | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $a \in S$.
Then $a$ is path-connected to itself. | Consider the constant mapping on the closed unit interval $\mathbb I = \left[{0 \,.\,.\, 1}\right]$:
:$\forall x \in \mathbb I: f_a \left({x}\right) = a$
Thus, in particular:
:$f_a \left({0}\right) = a$
:$f_a \left({1}\right) = a$
As a Constant Mapping is Continuous, it follows that $f_a$ is a path in $X$.
Thus $a$ is ... | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $a \in S$.
Then $a$ is [[Definition:Path-Connected Points|path-connected]] to itself. | Consider the [[Definition:Constant Mapping|constant mapping]] on the [[Definition:Closed Unit Interval|closed unit interval]] $\mathbb I = \left[{0 \,.\,.\, 1}\right]$:
:$\forall x \in \mathbb I: f_a \left({x}\right) = a$
Thus, in particular:
:$f_a \left({0}\right) = a$
:$f_a \left({1}\right) = a$
As a [[Constan... | Point is Path-Connected to Itself | https://proofwiki.org/wiki/Point_is_Path-Connected_to_Itself | https://proofwiki.org/wiki/Point_is_Path-Connected_to_Itself | [
"Path-Connected Spaces"
] | [
"Definition:Topological Space",
"Definition:Path-Connected/Points"
] | [
"Definition:Constant Mapping",
"Definition:Real Interval/Unit Interval/Closed",
"Constant Mapping is Continuous",
"Definition:Path (Topology)",
"Definition:Path-Connected/Points",
"Category:Path-Connected Spaces"
] |
proofwiki-3638 | Discrete Uniformity is Uniformity | Let $S$ be a set.
Let $\UU$ be the discrete uniformity on $S$
Then $\UU$ is indeed a uniformity. | We examine the uniformity axioms in turn:
;$\text U 1$: $\forall u \in \UU: \Delta_S \subseteq u$
This follows by definition of the discrete uniformity:
:$\UU := \set {u \subseteq S \times S: \Delta_S \subseteq u}$
;$\text U 2$: $\forall u, v \in \UU: u \cap v \in \UU$
We have that $\forall u, v \subseteq S \times S: \... | Let $S$ be a [[Definition:Set|set]].
Let $\UU$ be the [[Definition:Discrete Uniformity|discrete uniformity]] on $S$
Then $\UU$ is indeed a [[Definition:Uniformity|uniformity]]. | We examine the [[Axiom:Uniformity Axioms|uniformity axioms]] in turn:
;$\text U 1$: $\forall u \in \UU: \Delta_S \subseteq u$
This follows by definition of the [[Definition:Discrete Uniformity|discrete uniformity]]:
:$\UU := \set {u \subseteq S \times S: \Delta_S \subseteq u}$
;$\text U 2$: $\forall u, v \in \UU: ... | Discrete Uniformity is Uniformity | https://proofwiki.org/wiki/Discrete_Uniformity_is_Uniformity | https://proofwiki.org/wiki/Discrete_Uniformity_is_Uniformity | [
"Discrete Uniformities"
] | [
"Definition:Set",
"Definition:Discrete Uniformity",
"Definition:Uniformity"
] | [
"Axiom:Uniformity Axioms",
"Definition:Discrete Uniformity",
"Intersection is Largest Subset",
"Subset Relation is Transitive",
"Diagonal Relation is Equivalence",
"Definition:Equivalence Relation",
"Definition:Transitive Relation",
"Equivalence of Definitions of Transitive Relation",
"Axiom:Uniform... |
proofwiki-3639 | Discrete Uniformity generates Discrete Topology | Let $S$ be a set.
Let $\UU$ be the discrete uniformity on $S$.
Then the topology generated by $\UU$ is the discrete topology on $S$.
The diagonal relation $\Delta_S$ generates the basis for this discrete topology. | From the construction, let $\tau \subseteq \powerset S$ be a subset of the power set of $S$, created from $\UU$ by:
:$\tau := \set {\map u x: u \in \UU, x \in X}$
where:
:$\forall x \in X: \map u x = \set {y: \tuple {x, y} \in u}$
We need to show that $\tau$ is the discrete topology.
Consider $\Delta_S \in \UU$.
Let $x... | Let $S$ be a [[Definition:Set|set]].
Let $\UU$ be the [[Definition:Discrete Uniformity|discrete uniformity]] on $S$.
Then the [[Definition:Topology Induced by Quasiuniformity|topology generated by $\UU$]] is the [[Definition:Discrete Topology|discrete topology]] on $S$.
The [[Definition:Diagonal Relation|diagonal r... | From [[Definition:Topology Induced by Quasiuniformity|the construction]], let $\tau \subseteq \powerset S$ be a [[Definition:Subset|subset]] of the [[Definition:Power Set|power set]] of $S$, created from $\UU$ by:
:$\tau := \set {\map u x: u \in \UU, x \in X}$
where:
:$\forall x \in X: \map u x = \set {y: \tuple {x, y... | Discrete Uniformity generates Discrete Topology | https://proofwiki.org/wiki/Discrete_Uniformity_generates_Discrete_Topology | https://proofwiki.org/wiki/Discrete_Uniformity_generates_Discrete_Topology | [
"Discrete Uniformities",
"Discrete Topologies"
] | [
"Definition:Set",
"Definition:Discrete Uniformity",
"Definition:Topology Induced by Quasiuniformity",
"Definition:Discrete Topology",
"Definition:Diagonal Relation",
"Definition:Basis (Topology)",
"Definition:Discrete Topology"
] | [
"Definition:Topology Induced by Quasiuniformity",
"Definition:Subset",
"Definition:Power Set",
"Definition:Discrete Topology",
"Basis for Discrete Topology",
"Definition:Basis (Topology)",
"Definition:Discrete Topology"
] |
proofwiki-3640 | Indiscrete Topology is Topology | :$\tau$ is a topology on $S$. | Let $T = \struct {S, \set {\O, S} }$ be the indiscrete space on $S$.
Confirming the criteria for $T$ to be a topology:
:$(1): \quad$ Trivially, by definition, $\O \in \tau$ and $S \in \tau$.
:$(2): \quad \O \cup \O = \O \in \tau$, $\O \cup S = S \in \tau$ and $S \cup S = S \in \tau$ from Union with Empty Set and Set Un... | :$\tau$ is a [[Definition:Topology|topology]] on $S$. | Let $T = \struct {S, \set {\O, S} }$ be the [[Definition:Indiscrete Space|indiscrete space]] on $S$.
Confirming the criteria for $T$ to be a [[Definition:Topology|topology]]:
:$(1): \quad$ Trivially, by definition, $\O \in \tau$ and $S \in \tau$.
:$(2): \quad \O \cup \O = \O \in \tau$, $\O \cup S = S \in \tau$ and ... | Indiscrete Topology is Topology | https://proofwiki.org/wiki/Indiscrete_Topology_is_Topology | https://proofwiki.org/wiki/Indiscrete_Topology_is_Topology | [
"Indiscrete Topology"
] | [
"Definition:Topology"
] | [
"Definition:Indiscrete Topology",
"Definition:Topology",
"Union with Empty Set",
"Set Union is Idempotent",
"Intersection with Empty Set",
"Set Intersection is Idempotent"
] |
proofwiki-3641 | Indiscrete Topology is Coarsest Topology | :$\tau$ is the coarsest topology on $S$. | Let $\phi$ be any topology on $S$.
Then by definition of topology, $\O \in \phi$ and $S \in \phi$
Hence by definition of subset, $\tau \subseteq \phi$.
Hence by definition of coarser topology, $\tau$ is coarser than $\phi$.
{{qed}} | :$\tau$ is the [[Definition:Coarser Topology|coarsest topology]] on $S$. | Let $\phi$ be any [[Definition:Topology|topology]] on $S$.
Then by definition of [[Definition:Topology|topology]], $\O \in \phi$ and $S \in \phi$
Hence by definition of [[Definition:Subset|subset]], $\tau \subseteq \phi$.
Hence by definition of [[Definition:Coarser Topology|coarser topology]], $\tau$ is [[Definitio... | Indiscrete Topology is Coarsest Topology | https://proofwiki.org/wiki/Indiscrete_Topology_is_Coarsest_Topology | https://proofwiki.org/wiki/Indiscrete_Topology_is_Coarsest_Topology | [
"Indiscrete Topology",
"Examples of Coarser Topology"
] | [
"Definition:Coarser Topology"
] | [
"Definition:Topology",
"Definition:Topology",
"Definition:Subset",
"Definition:Coarser Topology",
"Definition:Coarser Topology"
] |
proofwiki-3642 | Limit Points of Indiscrete Space | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space consisting of at least two points.
Let $H$ be a subset of $T$ such that $H \ne \O$.
Then every point of $T$ is a limit point of $H$. | First we dispose of the case where $S$ is a singleton.
From Trivial Topological Space is Indiscrete, such a $T = \struct {S, \set {\O, S} }$ is a trivial topological space.
From Trivial Topological Space has no Limit Point, such a $T$ has no limit point.
{{Recall|Limit Point of Set/Definition from Open Neighborhood|lim... | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]] consisting of at least two [[Definition:Point of Set|points]].
Let $H$ be a [[Definition:Subset|subset]] of $T$ such that $H \ne \O$.
Then every [[Definition:Point of Set|point]] of $T$ is a [[Definition:Limit Poi... | First we dispose of the case where $S$ is a [[Definition:Singleton|singleton]].
From [[Trivial Topological Space is Indiscrete]], such a $T = \struct {S, \set {\O, S} }$ is a [[Definition:Trivial Topological Space|trivial topological space]].
From [[Trivial Topological Space has no Limit Point]], such a $T$ has no [[... | Limit Points of Indiscrete Space | https://proofwiki.org/wiki/Limit_Points_of_Indiscrete_Space | https://proofwiki.org/wiki/Limit_Points_of_Indiscrete_Space | [
"Indiscrete Topology",
"Examples of Limit Points"
] | [
"Definition:Indiscrete Topology",
"Definition:Element",
"Definition:Subset",
"Definition:Element",
"Definition:Limit Point/Topology/Set"
] | [
"Definition:Singleton",
"Trivial Topological Space is Indiscrete",
"Definition:Trivial Topological Space",
"Trivial Topological Space has no Limit Point",
"Definition:Limit Point/Topology/Set",
"Open Sets in Indiscrete Topology",
"Definition:Open Set/Topology",
"Definition:Element",
"Definition:Elem... |
proofwiki-3643 | Sequence in Indiscrete Space converges to Every Point | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Let $\sequence {s_n}$ be a sequence in $T$.
Then $\sequence {s_n}$ converges to every point of $S$. | Let $\alpha \in S$.
By definition, $\sequence {s_n}$ converges to $\alpha$ if every open set in $T$ containing $\alpha$ contains all but a finite number of terms of $\sequence {s_n}$.
But as $T$ has only one open set containing any points at all, '''every''' point of $\sequence {s_n}$ is contained in every open set in ... | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Let $\sequence {s_n}$ be a [[Definition:Sequence|sequence]] in $T$.
Then $\sequence {s_n}$ [[Definition:Convergent Sequence (Topology)|converges]] to every [[Definition:Point of Set|point]] of $S$. | Let $\alpha \in S$.
By definition, $\sequence {s_n}$ [[Definition:Convergent Sequence (Topology)|converges]] to $\alpha$ if every [[Definition:Open Set (Topology)|open set]] in $T$ containing $\alpha$ contains all but a [[Definition:Finite Set|finite number]] of terms of $\sequence {s_n}$.
But as $T$ has only one [[D... | Sequence in Indiscrete Space converges to Every Point | https://proofwiki.org/wiki/Sequence_in_Indiscrete_Space_converges_to_Every_Point | https://proofwiki.org/wiki/Sequence_in_Indiscrete_Space_converges_to_Every_Point | [
"Indiscrete Topology"
] | [
"Definition:Indiscrete Topology",
"Definition:Sequence",
"Definition:Convergent Sequence/Topology",
"Definition:Element"
] | [
"Definition:Convergent Sequence/Topology",
"Definition:Open Set/Topology",
"Definition:Finite Set",
"Definition:Open Set/Topology",
"Definition:Element",
"Definition:Open Set/Topology"
] |
proofwiki-3644 | Subset of Indiscrete Space is Dense-in-itself | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Let $H \subseteq S$ be a subset of $S$ containing more than one point.
Then $H$ is dense-in-itself. | Let $x \in H$.
Then as $H$ is not singleton, $\exists y \in H: y \ne x$.
Then every neighborhood of $x$ contains $y$, as the only open set of $T$ is $S$, which also contains both $x$ and $y$.
Hence $x$ is not isolated by definition.
$x$ is general, so all points in $H$ are similarly not isolated.
Hence the subset $H$ i... | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Let $H \subseteq S$ be a [[Definition:Subset|subset]] of $S$ containing more than one [[Definition:Point of Set|point]].
Then $H$ is [[Definition:Dense-in-itself|dense-in-itself]]. | Let $x \in H$.
Then as $H$ is not [[Definition:Singleton|singleton]], $\exists y \in H: y \ne x$.
Then every [[Definition:Neighborhood (Topology)|neighborhood]] of $x$ contains $y$, as the only [[Definition:Open Set (Topology)|open set]] of $T$ is $S$, which also contains both $x$ and $y$.
Hence $x$ is not [[Definit... | Subset of Indiscrete Space is Dense-in-itself | https://proofwiki.org/wiki/Subset_of_Indiscrete_Space_is_Dense-in-itself | https://proofwiki.org/wiki/Subset_of_Indiscrete_Space_is_Dense-in-itself | [
"Indiscrete Topology",
"Examples of Dense-in-itself"
] | [
"Definition:Indiscrete Topology",
"Definition:Subset",
"Definition:Element",
"Definition:Dense-in-itself"
] | [
"Definition:Singleton",
"Definition:Neighborhood (Topology)",
"Definition:Open Set/Topology",
"Definition:Isolated Point (Topology)",
"Definition:Isolated Point (Topology)",
"Definition:Dense-in-itself"
] |
proofwiki-3645 | Indiscrete Space is Non-Meager | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Then $T$ is non-meager. | Let $H \subseteq S$ such that $H \ne \O$.
From Limit Points of Indiscrete Space, every point of $S$ is a limit point of $H$.
So, by definition:
:$H^- = S$
where $H^-$ is the closure of $H$ in $T$.
Now the interior of $S$ is $S$ itself (trivially, by definition).
So:
:$\paren {H^-}^\circ = S \ne \O$
where $\paren {H^-}^... | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Then $T$ is [[Definition:Non-Meager Space|non-meager]]. | Let $H \subseteq S$ such that $H \ne \O$.
From [[Limit Points of Indiscrete Space]], every point of $S$ is a [[Definition:Limit Point of Set|limit point]] of $H$.
So, by definition:
:$H^- = S$
where $H^-$ is the [[Definition:Closure (Topology)|closure]] of $H$ in $T$.
Now the [[Definition:Interior (Topology)|interio... | Indiscrete Space is Non-Meager | https://proofwiki.org/wiki/Indiscrete_Space_is_Non-Meager | https://proofwiki.org/wiki/Indiscrete_Space_is_Non-Meager | [
"Indiscrete Topology",
"Examples of Non-Meager Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:Meager Space/Non-Meager"
] | [
"Limit Points of Indiscrete Space",
"Definition:Limit Point/Topology/Set",
"Definition:Closure (Topology)",
"Definition:Interior (Topology)",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)",
"Definition:Nowhere Dense",
"Empty Set is Nowhere Dense",
"Definition:Set Union/Countable U... |
proofwiki-3646 | Empty Set is Nowhere Dense | Let $T = \struct {S, \tau}$ be a topological space.
Then the empty set $\O$ is nowhere dense in $T$. | From Empty Set is Closed in Topological Space, $\O$ is closed in $T$.
From Closed Set Equals its Closure:
:$\O^- = \O$
where $\O^-$ is the the closure of $\O$.
From Empty Set is Element of Topology, $\O$ is open in $T$.
From the definition (trivially) we also have that:
:$\O^\circ = \O$
where $\O^\circ$ is the interior... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then the [[Definition:Empty Set|empty set]] $\O$ is [[Definition:Nowhere Dense|nowhere dense]] in $T$. | From [[Empty Set is Closed in Topological Space]], $\O$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
From [[Closed Set Equals its Closure]]:
:$\O^- = \O$
where $\O^-$ is the the [[Definition:Closure (Topology)|closure]] of $\O$.
From [[Empty Set is Element of Topology]], $\O$ is [[Definition:Open Set (Topol... | Empty Set is Nowhere Dense | https://proofwiki.org/wiki/Empty_Set_is_Nowhere_Dense | https://proofwiki.org/wiki/Empty_Set_is_Nowhere_Dense | [
"Empty Set",
"Examples of Nowhere Dense"
] | [
"Definition:Topological Space",
"Definition:Empty Set",
"Definition:Nowhere Dense"
] | [
"Empty Set is Closed/Topological Space",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topological Closure",
"Definition:Closure (Topology)",
"Empty Set is Element of Topology",
"Definition:Open Set/Topology",
"Definition:Interior (Topology)",
"Definition:Nowhere Dense"
] |
proofwiki-3647 | Interior of Open Set | Let $T = \left({S, \tau}\right)$ be a topological space.
Let $U \subseteq T$ be open in $T$.
Then:
: $U^\circ = U \iff U \in \tau$
where $U^\circ$ is the interior of $U$.
That is, a subset of $S$ is open in $T$ {{iff}} it equals its interior. | === Necessary Condition ===
Let $U \subseteq T$ be open in $T$.
From Set Interior is Largest Open Set:
: $U \subseteq U^\circ$
But by definition of interior:
: $U^\circ \subseteq U$
Hence $U^\circ = U$.
{{qed|lemma}} | Let $T = \left({S, \tau}\right)$ be a [[Definition:Topological Space|topological space]].
Let $U \subseteq T$ be [[Definition:Open Set (Topology)|open]] in $T$.
Then:
: $U^\circ = U \iff U \in \tau$
where $U^\circ$ is the [[Definition:Interior (Topology)|interior]] of $U$.
That is, a [[Definition:Subset|subset]] of... | === Necessary Condition ===
Let $U \subseteq T$ be [[Definition:Open Set (Topology)|open]] in $T$.
From [[Set Interior is Largest Open Set]]:
: $U \subseteq U^\circ$
But by definition of [[Definition:Interior (Topology)|interior]]:
: $U^\circ \subseteq U$
Hence $U^\circ = U$.
{{qed|lemma}} | Interior of Open Set | https://proofwiki.org/wiki/Interior_of_Open_Set | https://proofwiki.org/wiki/Interior_of_Open_Set | [
"Set Interiors",
"Open Sets"
] | [
"Definition:Topological Space",
"Definition:Open Set/Topology",
"Definition:Interior (Topology)",
"Definition:Subset",
"Definition:Open Set/Topology",
"Definition:Interior (Topology)"
] | [
"Definition:Open Set/Topology",
"Equivalence of Definitions of Interior (Topology)",
"Definition:Interior (Topology)",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology"
] |
proofwiki-3648 | Interior of Subset of Indiscrete Space | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Let $H \subsetneqq S$ (that is, let $H$ be a proper subset of $S$).
Then:
:$H^\circ = H^{\circ -} = H^{\circ - \circ} = \O$
where:
:$H^\circ$ denotes the interior of $H$
:$H^-$ denotes the closure of $H$. | As $H \subsetneqq S$, it follows that $H \ne S$.
So the only open subset of $H$ is $\O$.
So by definition:
:$H^\circ = \O$
From Empty Set is Closed in Topological Space, $\O$ is closed in $T$.
From Closed Set Equals its Closure:
:$\O^- = \O$
From Empty Set is Element of Topology, $\O$ is open in $T$.
From Interior of O... | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Let $H \subsetneqq S$ (that is, let $H$ be a [[Definition:Proper Subset|proper subset]] of $S$).
Then:
:$H^\circ = H^{\circ -} = H^{\circ - \circ} = \O$
where:
:$H^\circ$ denotes the [[Definition:Interior (Topolo... | As $H \subsetneqq S$, it follows that $H \ne S$.
So the only [[Definition:Open Set (Topology)|open subset]] of $H$ is $\O$.
So by definition:
:$H^\circ = \O$
From [[Empty Set is Closed in Topological Space]], $\O$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
From [[Closed Set Equals its Closure]]:
:$\O^- ... | Interior of Subset of Indiscrete Space | https://proofwiki.org/wiki/Interior_of_Subset_of_Indiscrete_Space | https://proofwiki.org/wiki/Interior_of_Subset_of_Indiscrete_Space | [
"Indiscrete Topology",
"Examples of Set Interiors"
] | [
"Definition:Indiscrete Topology",
"Definition:Proper Subset",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)"
] | [
"Definition:Open Set/Topology",
"Empty Set is Closed/Topological Space",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topological Closure",
"Empty Set is Element of Topology",
"Definition:Open Set/Topology",
"Interior of Open Set"
] |
proofwiki-3649 | Closure of Subset of Indiscrete Space | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Let $\O \subsetneq H \subseteq S$ (that is, let $H$ be a non-empty subset of $T$).
Then:
:$H^- = H^{- \circ} = H^{- \circ -} = S$
where:
:$H^\circ$ denotes the interior of $H$
:$H^-$ denotes the closure of $H$. | From Limit Points of Indiscrete Space, every point in $S$ is a limit point of $H$.
So from the definition of closure, $H \ne \O \implies H^-= S$.
From the open set axioms, $S$ is open in $T$.
From Interior of Open Set, $S^\circ = S$.
From Underlying Set of Topological Space is Clopen, $S$ is closed in $T$.
From Closed ... | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Let $\O \subsetneq H \subseteq S$ (that is, let $H$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Subset|subset]] of $T$).
Then:
:$H^- = H^{- \circ} = H^{- \circ -} = S$
where:
:$H^\circ$ denotes the [... | From [[Limit Points of Indiscrete Space]], every [[Definition:Point of Set|point]] in $S$ is a [[Definition:Limit Point of Set|limit point]] of $H$.
So from the definition of [[Definition:Closure (Topology)|closure]], $H \ne \O \implies H^-= S$.
From the [[Axiom:Open Set Axioms|open set axioms]], $S$ is [[Definition:... | Closure of Subset of Indiscrete Space | https://proofwiki.org/wiki/Closure_of_Subset_of_Indiscrete_Space | https://proofwiki.org/wiki/Closure_of_Subset_of_Indiscrete_Space | [
"Indiscrete Topology",
"Examples of Set Closures"
] | [
"Definition:Indiscrete Topology",
"Definition:Non-Empty Set",
"Definition:Subset",
"Definition:Interior (Topology)",
"Definition:Closure (Topology)"
] | [
"Limit Points of Indiscrete Space",
"Definition:Element",
"Definition:Limit Point/Topology/Set",
"Definition:Closure (Topology)",
"Axiom:Open Set Axioms",
"Definition:Open Set/Topology",
"Interior of Open Set",
"Underlying Set of Topological Space is Clopen",
"Definition:Closed Set/Topology",
"Set... |
proofwiki-3650 | Boundary of Subset of Indiscrete Space | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Let $\O \subsetneq H \subsetneq S$ (that is, let $H$ be a non-empty proper subset of $T$).
Then:
:$\partial H = S$
where $\partial H$ denotes the boundary of $H$.
If $H = \O$ or $H = S$ then $\partial H = \O$. | From Closure of Subset of Indiscrete Space, $H^- = S$, where $H^-$ denotes set closure.
From Interior of Subset of Indiscrete Space, $H^\circ = \O$, where $H^\circ$ denotes set interior.
By definition:
:$\partial H = H^- \setminus H^\circ = S \setminus \O = S$
From Open and Closed Sets in Topological Space, $\O$ and $S... | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Let $\O \subsetneq H \subsetneq S$ (that is, let $H$ be a [[Definition:Non-Empty Set|non-empty]] [[Definition:Proper Subset|proper subset]] of $T$).
Then:
:$\partial H = S$
where $\partial H$ denotes the [[Defini... | From [[Closure of Subset of Indiscrete Space]], $H^- = S$, where $H^-$ denotes [[Definition:Closure (Topology)|set closure]].
From [[Interior of Subset of Indiscrete Space]], $H^\circ = \O$, where $H^\circ$ denotes [[Definition:Interior (Topology)|set interior]].
By definition:
:$\partial H = H^- \setminus H^\circ = ... | Boundary of Subset of Indiscrete Space | https://proofwiki.org/wiki/Boundary_of_Subset_of_Indiscrete_Space | https://proofwiki.org/wiki/Boundary_of_Subset_of_Indiscrete_Space | [
"Indiscrete Topology",
"Examples of Set Boundaries"
] | [
"Definition:Indiscrete Topology",
"Definition:Non-Empty Set",
"Definition:Proper Subset",
"Definition:Boundary (Topology)"
] | [
"Closure of Subset of Indiscrete Space",
"Definition:Closure (Topology)",
"Interior of Subset of Indiscrete Space",
"Definition:Interior (Topology)",
"Open and Closed Sets in Topological Space",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Set is Clopen iff Boundary is Empty"
] |
proofwiki-3651 | Boundary of Boundary of Subset of Indiscrete Space | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Let $H \subseteq S$.
Then:
:$\map \partial {\partial H} = \O$
where $\partial H$ denotes the boundary of $H$. | From Boundary of Subset of Indiscrete Space, either $\partial H = S$ or $\partial H = \O$, depending on whether $H = \O$ or $H = S$ or not.
From Open and Closed Sets in Topological Space, $\O$ and $S$ are both closed and open in $T$.
So from Set Clopen iff Boundary is Empty:
:$\map \partial {\partial H} = \O$
{{qed}} | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Let $H \subseteq S$.
Then:
:$\map \partial {\partial H} = \O$
where $\partial H$ denotes the [[Definition:Boundary (Topology)|boundary]] of $H$. | From [[Boundary of Subset of Indiscrete Space]], either $\partial H = S$ or $\partial H = \O$, depending on whether $H = \O$ or $H = S$ or not.
From [[Open and Closed Sets in Topological Space]], $\O$ and $S$ are both [[Definition:Closed Set (Topology)|closed]] and [[Definition:Open Set (Topology)|open]] in $T$.
So f... | Boundary of Boundary of Subset of Indiscrete Space | https://proofwiki.org/wiki/Boundary_of_Boundary_of_Subset_of_Indiscrete_Space | https://proofwiki.org/wiki/Boundary_of_Boundary_of_Subset_of_Indiscrete_Space | [
"Indiscrete Topology",
"Examples of Set Boundaries"
] | [
"Definition:Indiscrete Topology",
"Definition:Boundary (Topology)"
] | [
"Boundary of Subset of Indiscrete Space",
"Open and Closed Sets in Topological Space",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Set is Clopen iff Boundary is Empty"
] |
proofwiki-3652 | Subset of Indiscrete Space is Everywhere Dense | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Let $H \subseteq S$ such that $H \ne \O$.
Then $H$ is everywhere dense. | From Limit Points of Indiscrete Space, every point of $T$ is a limit point of $H$.
Hence $H$ is everywhere dense by definition.
{{qed}} | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Let $H \subseteq S$ such that $H \ne \O$.
Then $H$ is [[Definition:Everywhere Dense|everywhere dense]]. | From [[Limit Points of Indiscrete Space]], every point of $T$ is a [[Definition:Limit Point of Set|limit point]] of $H$.
Hence $H$ is [[Definition:Everywhere Dense|everywhere dense]] by definition.
{{qed}} | Subset of Indiscrete Space is Everywhere Dense | https://proofwiki.org/wiki/Subset_of_Indiscrete_Space_is_Everywhere_Dense | https://proofwiki.org/wiki/Subset_of_Indiscrete_Space_is_Everywhere_Dense | [
"Indiscrete Topology",
"Examples of Everywhere Dense"
] | [
"Definition:Indiscrete Topology",
"Definition:Everywhere Dense"
] | [
"Limit Points of Indiscrete Space",
"Definition:Limit Point/Topology/Set",
"Definition:Everywhere Dense"
] |
proofwiki-3653 | Indiscrete Space is Second-Countable | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Then $T$ is a second-countable space. | The only basis for $T$ is $\set S$ which is trivially countable.
Hence the result.
{{qed}} | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Then $T$ is a [[Definition:Second-Countable Space|second-countable space]]. | The only [[Definition:Basis (Topology)|basis]] for $T$ is $\set S$ which is trivially [[Definition:Countable Set|countable]].
Hence the result.
{{qed}} | Indiscrete Space is Second-Countable | https://proofwiki.org/wiki/Indiscrete_Space_is_Second-Countable | https://proofwiki.org/wiki/Indiscrete_Space_is_Second-Countable | [
"Indiscrete Topology",
"Examples of Second-Countable Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:Second-Countable Space"
] | [
"Definition:Basis (Topology)",
"Definition:Countable Set"
] |
proofwiki-3654 | Mapping to Indiscrete Space is Continuous | Let $T_1 = \struct {S_1, \tau_1}$ be any topological space.
Let $T_2 = \struct {S_2, \tau_2}$ be the indiscrete topological space on $S_2$.
Let $\phi: S_1 \to S_2$ be a mapping.
Then $\phi$ is continuous. | From the definition of continuous:
:$U \in \tau_2 \implies \phi^{-1} \sqbrk U \in \tau_1$
The only elements of $\tau_2$ are $S_2$ and $\O$, from which:
:$\phi^{-1} \sqbrk {S_2} = S_1 \in \tau_1$
:$\phi^{-1} \sqbrk \O = \O \in \tau_1$
{{qed}} | Let $T_1 = \struct {S_1, \tau_1}$ be any [[Definition:Topological Space|topological space]].
Let $T_2 = \struct {S_2, \tau_2}$ be the [[Definition:Indiscrete Space|indiscrete topological space]] on $S_2$.
Let $\phi: S_1 \to S_2$ be a [[Definition:Mapping|mapping]].
Then $\phi$ is [[Definition:Everywhere Continuous ... | From the definition of [[Definition:Everywhere Continuous Mapping (Topology)|continuous]]:
:$U \in \tau_2 \implies \phi^{-1} \sqbrk U \in \tau_1$
The only [[Definition:Element|elements]] of $\tau_2$ are $S_2$ and $\O$, from which:
:$\phi^{-1} \sqbrk {S_2} = S_1 \in \tau_1$
:$\phi^{-1} \sqbrk \O = \O \in \tau_1$
{{qe... | Mapping to Indiscrete Space is Continuous | https://proofwiki.org/wiki/Mapping_to_Indiscrete_Space_is_Continuous | https://proofwiki.org/wiki/Mapping_to_Indiscrete_Space_is_Continuous | [
"Indiscrete Topology",
"Continuous Mappings (Topology)"
] | [
"Definition:Topological Space",
"Definition:Indiscrete Topology",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)/Everywhere"
] | [
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Element"
] |
proofwiki-3655 | Indiscrete Space is Path-Connected | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Then $T$ is path-connected. | Let $a, b \in S$.
Consider any mapping $f: \closedint 0 1 \to S$ such that $\map f 0 = a$ and $\map f 1 = b$.
From Mapping to Indiscrete Space is Continuous, we have that $f$ is continuous.
The result follows by definition of path-connectedness.
{{qed}} | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Then $T$ is [[Definition:Path-Connected Space|path-connected]]. | Let $a, b \in S$.
Consider any [[Definition:Mapping|mapping]] $f: \closedint 0 1 \to S$ such that $\map f 0 = a$ and $\map f 1 = b$.
From [[Mapping to Indiscrete Space is Continuous]], we have that $f$ is [[Definition:Everywhere Continuous Mapping (Topology)|continuous]].
The result follows by definition of [[Defini... | Indiscrete Space is Path-Connected | https://proofwiki.org/wiki/Indiscrete_Space_is_Path-Connected | https://proofwiki.org/wiki/Indiscrete_Space_is_Path-Connected | [
"Indiscrete Topology",
"Examples of Path-Connected Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:Path-Connected/Topological Space"
] | [
"Definition:Mapping",
"Mapping to Indiscrete Space is Continuous",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Path-Connected/Topological Space"
] |
proofwiki-3656 | Indiscrete Space is Connected | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Then $T$ is connected. | We have that an Indiscrete Space is Path-Connected.
Then we have that a Path-Connected Space is Connected.
{{qed}} | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Then $T$ is [[Definition:Connected Topological Space|connected]]. | We have that an [[Indiscrete Space is Path-Connected]].
Then we have that a [[Path-Connected Space is Connected]].
{{qed}} | Indiscrete Space is Connected | https://proofwiki.org/wiki/Indiscrete_Space_is_Connected | https://proofwiki.org/wiki/Indiscrete_Space_is_Connected | [
"Indiscrete Topology",
"Examples of Connected Topological Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:Connected Topological Space"
] | [
"Indiscrete Space is Path-Connected",
"Path-Connected Space is Connected"
] |
proofwiki-3657 | Indiscrete Space is Injectively Path-Connected iff Uncountable | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Then $T$ is injectively path-connected {{iff}} $S$ is an uncountable set. | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space such that $S$ is uncountable.
Let $a, b \in S$.
Consider an injection $f: \closedint 0 1 \to S$ such that $\map f 0 = a$ and $\map f 1 = b$.
This can always be found because $S$ is itself uncountable.
From Mapping to Indiscrete Space is Continuous,... | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Then $T$ is [[Definition:Injectively Path-Connected Space|injectively path-connected]] {{iff}} $S$ is an [[Definition:Uncountable Set|uncountable set]]. | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]] such that $S$ is [[Definition:Uncountable Set|uncountable]].
Let $a, b \in S$.
Consider an [[Definition:Injection|injection]] $f: \closedint 0 1 \to S$ such that $\map f 0 = a$ and $\map f 1 = b$.
This can always... | Indiscrete Space is Injectively Path-Connected iff Uncountable | https://proofwiki.org/wiki/Indiscrete_Space_is_Injectively_Path-Connected_iff_Uncountable | https://proofwiki.org/wiki/Indiscrete_Space_is_Injectively_Path-Connected_iff_Uncountable | [
"Indiscrete Topology",
"Examples of Injectively Path-Connected Spaces",
"Uncountable Sets"
] | [
"Definition:Indiscrete Topology",
"Definition:Injectively Path-Connected/Topological Space",
"Definition:Uncountable/Set"
] | [
"Definition:Indiscrete Topology",
"Definition:Uncountable/Set",
"Definition:Injection",
"Definition:Uncountable/Set",
"Mapping to Indiscrete Space is Continuous",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Injectively Path-Connected/Topological Space",
"Definition:Indiscrete To... |
proofwiki-3658 | Indiscrete Space is Irreducible | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Then $T$ is irreducible. | There is only one non-empty open set in $T$.
So there can be no two open sets in $T$ which are disjoint.
Hence (trivially) $T$ is irreducible.
{{qed}} | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Then $T$ is [[Definition:Irreducible Space|irreducible]]. | There is only one [[Definition:Non-Empty Set|non-empty]] [[Definition:Open Set (Topology)|open set]] in $T$.
So there can be no two [[Definition:Open Set (Topology)|open sets]] in $T$ which are [[Definition:Disjoint Sets|disjoint]].
Hence (trivially) $T$ is [[Definition:Irreducible Space|irreducible]].
{{qed}} | Indiscrete Space is Irreducible | https://proofwiki.org/wiki/Indiscrete_Space_is_Irreducible | https://proofwiki.org/wiki/Indiscrete_Space_is_Irreducible | [
"Indiscrete Topology",
"Examples of Irreducible Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:Irreducible Space"
] | [
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Definition:Disjoint Sets",
"Definition:Irreducible Space"
] |
proofwiki-3659 | Indiscrete Space is Ultraconnected | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Then $T$ is ultraconnected. | There is only one non-empty closed set in $T$.
So there can be no two closed sets in $T$ which are disjoint.
Hence (trivially) $T$ is ultraconnected.
{{qed}} | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Then $T$ is [[Definition:Ultraconnected Space|ultraconnected]]. | There is only one [[Definition:Non-Empty Set|non-empty]] [[Definition:Closed Set (Topology)|closed set]] in $T$.
So there can be no two [[Definition:Closed Set (Topology)|closed sets]] in $T$ which are [[Definition:Disjoint Sets|disjoint]].
Hence (trivially) $T$ is [[Definition:Ultraconnected Space|ultraconnected]].
... | Indiscrete Space is Ultraconnected | https://proofwiki.org/wiki/Indiscrete_Space_is_Ultraconnected | https://proofwiki.org/wiki/Indiscrete_Space_is_Ultraconnected | [
"Indiscrete Topology",
"Examples of Ultraconnected Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:Ultraconnected Space"
] | [
"Definition:Non-Empty Set",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Disjoint Sets",
"Definition:Ultraconnected Space"
] |
proofwiki-3660 | Indiscrete Non-Singleton Space is not T0 | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space which has more than one element.
Then $T$ is not a $T_0$ space. | Let $a, b \in S$.
By definition of indiscrete space, $S$ is the only open set in $T$ which is non-empty.
So (trivially) there is no open set in $T$ containing $a$ and not $b$, or $b$ and not $a$.
Hence the result, by definition of $T_0$ space.
{{qed}} | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]] which has more than one [[Definition:Element|element]].
Then $T$ is not a [[Definition:T0 Space|$T_0$ space]]. | Let $a, b \in S$.
By definition of [[Definition:Indiscrete Space|indiscrete space]], $S$ is the only [[Definition:Open Set (Topology)|open set]] in $T$ which is [[Definition:Non-Empty Set|non-empty]].
So (trivially) there is no [[Definition:Open Set (Topology)|open set]] in $T$ containing $a$ and not $b$, or $b$ and ... | Indiscrete Non-Singleton Space is not T0 | https://proofwiki.org/wiki/Indiscrete_Non-Singleton_Space_is_not_T0 | https://proofwiki.org/wiki/Indiscrete_Non-Singleton_Space_is_not_T0 | [
"Indiscrete Topology",
"Examples of T0 Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:Element",
"Definition:T0 Space"
] | [
"Definition:Indiscrete Topology",
"Definition:Open Set/Topology",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Definition:T0 Space"
] |
proofwiki-3661 | Indiscrete Space is T5 | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Then $T$ is a $T_5$ space. | By definition, the only two separated sets in $T$ are $S$ and $\O$.
Then there exist two disjoint open sets $S$ and $\O$ containing $S$ and $\O$ respectively.
Hence (trivially) $T$ is a $T_5$ space.
{{qed}} | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Then $T$ is a [[Definition:T5 Space|$T_5$ space]]. | By definition, the only two [[Definition:Separated Sets|separated sets]] in $T$ are $S$ and $\O$.
Then there exist two [[Definition:Disjoint Sets|disjoint]] [[Definition:Open Set (Topology)|open sets]] $S$ and $\O$ containing $S$ and $\O$ respectively.
Hence (trivially) $T$ is a [[Definition:T5 Space|$T_5$ space]].
{... | Indiscrete Space is T5 | https://proofwiki.org/wiki/Indiscrete_Space_is_T5 | https://proofwiki.org/wiki/Indiscrete_Space_is_T5 | [
"Indiscrete Topology",
"Examples of T5 Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:T5 Space"
] | [
"Definition:Separated Sets",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology",
"Definition:T5 Space"
] |
proofwiki-3662 | Indiscrete Space is T4 | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Then $T$ is a $T_4$ space. | We have that an indiscrete space is a $T_5$ space.
Then we have that a $T_5$ Space is $T_4$.
{{qed}} | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Then $T$ is a [[Definition:T4 Space|$T_4$ space]]. | We have that an [[Indiscrete Space is T5|indiscrete space is a $T_5$ space]].
Then we have that a [[T5 Space is T4|$T_5$ Space is $T_4$]].
{{qed}} | Indiscrete Space is T4 | https://proofwiki.org/wiki/Indiscrete_Space_is_T4 | https://proofwiki.org/wiki/Indiscrete_Space_is_T4 | [
"Indiscrete Topology",
"Examples of T4 Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:T4 Space"
] | [
"Indiscrete Space is T5",
"T5 Space is T4"
] |
proofwiki-3663 | Indiscrete Space is T3 | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space .
Then $T$ is a $T_3$ space. | Let $F \subseteq S$ be a closed set in $S$.
Let $y \in S$ such that $y \notin F$.
The only way this can happen is if $F = \O$.
So there exist disjoint open sets $U, V \in \tau$ such that $F \subseteq U$, $y \in V$.
That is, $U = \O$ and $V = S$.
Hence (trivially) the result.
{{qed}} | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]] .
Then $T$ is a [[Definition:T3 Space|$T_3$ space]]. | Let $F \subseteq S$ be a [[Definition:Closed Set (Topology)|closed set]] in $S$.
Let $y \in S$ such that $y \notin F$.
The only way this can happen is if $F = \O$.
So there exist [[Definition:Disjoint Sets|disjoint]] [[Definition:Open Set (Topology)|open sets]] $U, V \in \tau$ such that $F \subseteq U$, $y \in V$.
... | Indiscrete Space is T3 | https://proofwiki.org/wiki/Indiscrete_Space_is_T3 | https://proofwiki.org/wiki/Indiscrete_Space_is_T3 | [
"Indiscrete Topology",
"Examples of T3 Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:T3 Space"
] | [
"Definition:Closed Set/Topology",
"Definition:Disjoint Sets",
"Definition:Open Set/Topology"
] |
proofwiki-3664 | Indiscrete Space is Pseudometrizable | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space.
Then $T$ is pseudometrizable. | Let $d: S \times S \to \R$ be the mapping defined as:
:$\forall x, y \in S: \map d {x, y} = 0$
Then clearly $d$ is a pseudometric.
Let $\struct {S, \tau_{\struct {S, d} } }$ be the topological space induced by $d$.
Since $\struct {S, \tau_{\struct {S, d} } }$ is a topological space, by Empty Set is Element of Topology ... | Let $T = \struct {S, \set {\O, S} }$ be an [[Definition:Indiscrete Space|indiscrete topological space]].
Then $T$ is [[Definition:Pseudometrizable Topology|pseudometrizable]]. | Let $d: S \times S \to \R$ be the [[Definition:Mapping|mapping]] defined as:
:$\forall x, y \in S: \map d {x, y} = 0$
Then clearly $d$ is a [[Definition:Pseudometric|pseudometric]].
Let $\struct {S, \tau_{\struct {S, d} } }$ be the [[Pseudometric induces Topology|topological space induced]] by $d$.
Since $\struct ... | Indiscrete Space is Pseudometrizable | https://proofwiki.org/wiki/Indiscrete_Space_is_Pseudometrizable | https://proofwiki.org/wiki/Indiscrete_Space_is_Pseudometrizable | [
"Indiscrete Topology",
"Examples of Pseudometrizable Topologies"
] | [
"Definition:Indiscrete Topology",
"Definition:Pseudometrizable Topology"
] | [
"Definition:Mapping",
"Definition:Pseudometric",
"Pseudometric induces Topology",
"Definition:Topological Space",
"Empty Set is Element of Topology",
"Definition:Open Set/Topology",
"Definition:Open Ball",
"Definition:Non-Empty Set",
"Definition:Open Set/Topology",
"Open Sets in Pseudometric Space... |
proofwiki-3665 | Partition Topology is Topology | Let $S$ be a set.
Let $\PP$ be a partition of $S$.
Let $\tau$ be the set of subsets of $S$ defined as:
:$a \in \tau \iff a$ is the union of sets of $\PP$
Then $\tau$ is a topology on $S$. | From Basis for Partition Topology, we have that $\PP$ is a basis for the partition topology.
The result follows.
{{qed}} | Let $S$ be a [[Definition:Set|set]].
Let $\PP$ be a [[Definition:Partition (Set Theory)|partition]] of $S$.
Let $\tau$ be the [[Definition:Set of Sets|set of subsets]] of $S$ defined as:
:$a \in \tau \iff a$ is the [[Definition:Set Union|union]] of sets of $\PP$
Then $\tau$ is a [[Definition:Topology|topology]] on ... | From [[Basis for Partition Topology]], we have that $\PP$ is a [[Definition:Basis (Topology)|basis]] for the [[Definition:Partition Topology|partition topology]].
The result follows.
{{qed}} | Partition Topology is Topology | https://proofwiki.org/wiki/Partition_Topology_is_Topology | https://proofwiki.org/wiki/Partition_Topology_is_Topology | [
"Partition Topologies"
] | [
"Definition:Set",
"Definition:Set Partition",
"Definition:Set of Sets",
"Definition:Set Union",
"Definition:Topology"
] | [
"Basis for Partition Topology",
"Definition:Basis (Topology)",
"Definition:Partition Topology"
] |
proofwiki-3666 | Basis for Partition Topology | Let $S$ be a set.
Let $\PP$ be a partition of $S$.
Let $\tau$ be the partition topology on $S$ defined as:
:$a \in \tau \iff a$ is the union of sets of $\PP$
Then $\PP$ forms a basis of $\tau$. | Checking the criteria for $\PP$ to be a synthetic basis for $\tau$:
We have that $\ds S = \bigcup \PP$ from the definition of a partition.
Therefore, $\ds S \subseteq \bigcup \PP$ and $\PP$ is a cover for $S$.
Next, let $B_1, B_2 \in \PP$.
Then as $\PP$ is a partition of $S$, we have that $B_1 \cap B_2 = \O$.
But from ... | Let $S$ be a [[Definition:Set|set]].
Let $\PP$ be a [[Definition:Partition (Set Theory)|partition]] of $S$.
Let $\tau$ be the [[Definition:Partition Topology|partition topology]] on $S$ defined as:
:$a \in \tau \iff a$ is the [[Definition:Set Union|union]] of [[Definition:Set|sets]] of $\PP$
Then $\PP$ forms a [[De... | Checking the criteria for $\PP$ to be a [[Definition:Synthetic Basis|synthetic basis]] for $\tau$:
We have that $\ds S = \bigcup \PP$ from the definition of a [[Definition:Partition (Set Theory)|partition]].
Therefore, $\ds S \subseteq \bigcup \PP$ and $\PP$ is a [[Definition:Cover of Set|cover]] for $S$.
Next, le... | Basis for Partition Topology | https://proofwiki.org/wiki/Basis_for_Partition_Topology | https://proofwiki.org/wiki/Basis_for_Partition_Topology | [
"Partition Topologies",
"Examples of Topological Bases"
] | [
"Definition:Set",
"Definition:Set Partition",
"Definition:Partition Topology",
"Definition:Set Union",
"Definition:Set",
"Definition:Basis (Topology)"
] | [
"Definition:Basis (Topology)/Synthetic Basis",
"Definition:Set Partition",
"Definition:Cover of Set",
"Definition:Set Partition",
"Union of Empty Set",
"Definition:Set Union",
"Definition:Set",
"Definition:Basis (Topology)/Synthetic Basis"
] |
proofwiki-3667 | Subset of Partition Space is Open iff Closed | Let $T = \struct {S, \tau}$ be a topological space.
Then $T$ is a partition space {{iff}}:
:$\forall U \subseteq S: U \in \tau \iff \relcomp S U \in \tau$
That is, a topological space is a partition space {{iff}}:
:all open sets are closed
:all closed sets are also open.
That is, all open sets and closed sets of a part... | Let $T = \struct {S, \tau}$ be a topological space. | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]].
Then $T$ is a [[Definition:Partition Space|partition space]] {{iff}}:
:$\forall U \subseteq S: U \in \tau \iff \relcomp S U \in \tau$
That is, a [[Definition:Topological Space|topological space]] is a [[Definition:Partition Space|par... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]]. | Subset of Partition Space is Open iff Closed | https://proofwiki.org/wiki/Subset_of_Partition_Space_is_Open_iff_Closed | https://proofwiki.org/wiki/Subset_of_Partition_Space_is_Open_iff_Closed | [
"Partition Topologies",
"Examples of Clopen Sets"
] | [
"Definition:Topological Space",
"Definition:Partition Topology",
"Definition:Topological Space",
"Definition:Partition Topology",
"Definition:Open Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Open Set/Topology",
"Def... | [
"Definition:Topological Space"
] |
proofwiki-3668 | Open Set in Partition Topology is Component | Let $T = \struct {S, \tau}$ be a partition topological space.
Then each of its open sets is a component of $T$. | Let the partition $\PP$ be a basis of $T$.
From Subset of Partition Space is Open iff Closed, open sets are in fact clopen.
So the elements of $\PP$ are clopen.
The result follows from the definition of components.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Partition Space|partition topological space]].
Then each of its [[Definition:Open Set (Topology)|open sets]] is a [[Definition:Component (Topology)|component]] of $T$. | Let the [[Definition:Partition (Topology)|partition]] $\PP$ be a [[Definition:Basis (Topology)|basis]] of $T$.
From [[Subset of Partition Space is Open iff Closed]], [[Definition:Open Set (Topology)|open sets]] are in fact [[Definition:Clopen Set|clopen]].
So the [[Definition:Element|elements]] of $\PP$ are [[Definit... | Open Set in Partition Topology is Component | https://proofwiki.org/wiki/Open_Set_in_Partition_Topology_is_Component | https://proofwiki.org/wiki/Open_Set_in_Partition_Topology_is_Component | [
"Partition Topologies",
"Examples of Topological Components"
] | [
"Definition:Partition Topology",
"Definition:Open Set/Topology",
"Definition:Component (Topology)"
] | [
"Definition:Separation (Topology)",
"Definition:Basis (Topology)",
"Subset of Partition Space is Open iff Closed",
"Definition:Open Set/Topology",
"Definition:Clopen Set",
"Definition:Element",
"Definition:Clopen Set",
"Definition:Component (Topology)"
] |
proofwiki-3669 | Quotient Topology of Partition Topology is Discrete Space | Let $\PP$ be a partition of a set $S$.
Let $T = \struct {S, \tau}$ be the partition space formed from $\PP$.
Let $S / \PP$ be the quotient set of $S$ by $\PP$.
Then the quotient topology $\tau_{S / \PP}$ is a discrete topology. | Let $\BB$ be the set defined as:
:$\BB = \set {\set A: A \in S / \PP}$
From Basis for Partition Topology, $\BB$ forms a basis for a partition space on $S$.
From Basis for Discrete Topology, $\BB$ forms a basis for the discrete topology on $S / \PP$.
Hence the result, by definition of quotient topology.
{{qed}} | Let $\PP$ be a [[Definition:Partition (Set Theory)|partition]] of a [[Definition:Set|set]] $S$.
Let $T = \struct {S, \tau}$ be the [[Definition:Partition Space|partition space]] formed from $\PP$.
Let $S / \PP$ be the [[Definition:Quotient Set|quotient set]] of $S$ by $\PP$.
Then the [[Definition:Quotient Topology|... | Let $\BB$ be the [[Definition:Set|set]] defined as:
:$\BB = \set {\set A: A \in S / \PP}$
From [[Basis for Partition Topology]], $\BB$ forms a [[Definition:Basis (Topology)|basis]] for a [[Definition:Partition Space|partition space]] on $S$.
From [[Basis for Discrete Topology]], $\BB$ forms a [[Definition:Basis (Topo... | Quotient Topology of Partition Topology is Discrete Space | https://proofwiki.org/wiki/Quotient_Topology_of_Partition_Topology_is_Discrete_Space | https://proofwiki.org/wiki/Quotient_Topology_of_Partition_Topology_is_Discrete_Space | [
"Partition Topologies",
"Discrete Topologies",
"Examples of Quotient Topologies"
] | [
"Definition:Set Partition",
"Definition:Set",
"Definition:Partition Topology",
"Definition:Quotient Set",
"Definition:Quotient Topology",
"Definition:Discrete Topology"
] | [
"Definition:Set",
"Basis for Partition Topology",
"Definition:Basis (Topology)",
"Definition:Partition Topology",
"Basis for Discrete Topology",
"Definition:Basis (Topology)",
"Definition:Discrete Topology",
"Definition:Quotient Topology"
] |
proofwiki-3670 | Singleton Partition yields Indiscrete Topology | Let $S$ be a set which is not empty.
Let $\PP$ be the (trivial) singleton partition $\set S$ on $S$.
Then the partition topology on $\PP$ is the indiscrete topology. | By definition, the partition topology on $\PP$ is the set of all unions from $\PP$.
This is (trivially, and from Union of Empty Set) $\set {\O, S}$ which is the indiscrete topology on $S$ by definition.
{{qed}} | Let $S$ be a [[Definition:Set|set]] which is [[Definition:Non-Empty Set|not empty]].
Let $\PP$ be the [[Definition:Singleton Partition|(trivial) singleton partition]] $\set S$ on $S$.
Then the [[Definition:Partition Topology|partition topology]] on $\PP$ is the [[Definition:Indiscrete Topology|indiscrete topology]]. | By definition, the [[Definition:Partition Topology|partition topology]] on $\PP$ is the set of all [[Definition:Set Union|unions]] from $\PP$.
This is (trivially, and from [[Union of Empty Set]]) $\set {\O, S}$ which is the [[Definition:Indiscrete Topology|indiscrete topology]] on $S$ by definition.
{{qed}} | Singleton Partition yields Indiscrete Topology | https://proofwiki.org/wiki/Singleton_Partition_yields_Indiscrete_Topology | https://proofwiki.org/wiki/Singleton_Partition_yields_Indiscrete_Topology | [
"Indiscrete Topology",
"Partition Topologies"
] | [
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Trivial Partition/Singleton",
"Definition:Partition Topology",
"Definition:Indiscrete Topology"
] | [
"Definition:Partition Topology",
"Definition:Set Union",
"Union of Empty Set",
"Definition:Indiscrete Topology"
] |
proofwiki-3671 | Partition of Singletons yields Discrete Topology | Let $S$ be a set which is non-empty.
Let $\PP$ be the (trivial) partition of singletons on $S$:
:$\PP = \set {\set x: x \in S}$
Then the partition topology on $\PP$ is the discrete topology. | From Basis for Discrete Topology it is shown that $\PP$ as defined here forms the basis of the discrete topology.
{{qed}} | Let $S$ be a [[Definition:Set|set]] which is [[Definition:Non-Empty Set|non-empty]].
Let $\PP$ be the [[Definition:Partition of Singletons|(trivial) partition of singletons]] on $S$:
:$\PP = \set {\set x: x \in S}$
Then the [[Definition:Partition Topology|partition topology]] on $\PP$ is the [[Definition:Discrete To... | From [[Basis for Discrete Topology]] it is shown that $\PP$ as defined here forms the [[Definition:Basis (Topology)|basis]] of the [[Definition:Discrete Topology|discrete topology]].
{{qed}} | Partition of Singletons yields Discrete Topology | https://proofwiki.org/wiki/Partition_of_Singletons_yields_Discrete_Topology | https://proofwiki.org/wiki/Partition_of_Singletons_yields_Discrete_Topology | [
"Discrete Topologies",
"Partition Topologies"
] | [
"Definition:Set",
"Definition:Non-Empty Set",
"Definition:Trivial Partition/Partition of Singletons",
"Definition:Partition Topology",
"Definition:Discrete Topology"
] | [
"Basis for Discrete Topology",
"Definition:Basis (Topology)",
"Definition:Discrete Topology"
] |
proofwiki-3672 | Partition Topology is not T0 | Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then $T$ is not a $T_0$ space. | As $\PP$ is not the partition of singletons, there exists some $H \in \PP$ such that $a, b \in H: a \ne b$.
Any union of sets from $\PP$ which includes $H$ will therefore contain both $a$ and $b$.
Therefore, any element of $\tau$ containing $a$ will also contain $b$, and similarly, any element of $\tau$ containing $b$ ... | Let $S$ be a [[Definition:Set|set]] and let $\PP$ be a [[Definition:Partition (Set Theory)|partition]] on $S$ which is not the [[Definition:Partition of Singletons|(trivial) partition of singletons]].
Let $T = \struct {S, \tau}$ be the [[Definition:Partition Space|partition space]] whose [[Basis for Partition Topology... | As $\PP$ is not the [[Definition:Partition of Singletons|partition of singletons]], there exists some $H \in \PP$ such that $a, b \in H: a \ne b$.
Any [[Definition:Set Union|union]] of sets from $\PP$ which includes $H$ will therefore contain both $a$ and $b$.
Therefore, any element of $\tau$ containing $a$ will also... | Partition Topology is not T0 | https://proofwiki.org/wiki/Partition_Topology_is_not_T0 | https://proofwiki.org/wiki/Partition_Topology_is_not_T0 | [
"Partition Topologies",
"Examples of T0 Spaces"
] | [
"Definition:Set",
"Definition:Set Partition",
"Definition:Trivial Partition/Partition of Singletons",
"Definition:Partition Topology",
"Basis for Partition Topology",
"Definition:T0 Space"
] | [
"Definition:Trivial Partition/Partition of Singletons",
"Definition:Set Union",
"Definition:Open Set/Topology",
"Definition:T0 Space",
"Definition:Trivial Partition/Partition of Singletons",
"Partition of Singletons yields Discrete Topology",
"Discrete Space satisfies all Separation Properties",
"Defi... |
proofwiki-3673 | Partition Topology is T5 | Let $S$ be a set and let $\PP$ be a partition on $S$ which is not the (trivial) partition of singletons.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then $T$ is a $T_5$ space. | Let $A$ and $B$ be subsets of $S$ such that $A^- \cap B = A \cap B^- = \O$.
From Subset of Partition Space is Open iff Closed, we get that $A^-$ and $B^-$ are both clopen.
From Set is Subset of its Topological Closure:
:$A \subseteq A^-$
From $A^- \cap B = \O$ it follows from Intersection with Complement is Empty iff ... | Let $S$ be a [[Definition:Set|set]] and let $\PP$ be a [[Definition:Partition (Set Theory)|partition]] on $S$ which is not the [[Definition:Partition of Singletons|(trivial) partition of singletons]].
Let $T = \struct {S, \tau}$ be the [[Definition:Partition Space|partition space]] whose [[Basis for Partition Topology... | Let $A$ and $B$ be [[Definition:Subset|subsets]] of $S$ such that $A^- \cap B = A \cap B^- = \O$.
From [[Subset of Partition Space is Open iff Closed]], we get that $A^-$ and $B^-$ are both [[Definition:Clopen Set|clopen]].
From [[Set is Subset of its Topological Closure]]:
:$A \subseteq A^-$
From $A^- \cap B = \O$... | Partition Topology is T5 | https://proofwiki.org/wiki/Partition_Topology_is_T5 | https://proofwiki.org/wiki/Partition_Topology_is_T5 | [
"Partition Topologies",
"Examples of T5 Spaces"
] | [
"Definition:Set",
"Definition:Set Partition",
"Definition:Trivial Partition/Partition of Singletons",
"Definition:Partition Topology",
"Basis for Partition Topology",
"Definition:T5 Space"
] | [
"Definition:Subset",
"Subset of Partition Space is Open iff Closed",
"Definition:Clopen Set",
"Set is Subset of its Topological Closure",
"Intersection with Complement is Empty iff Subset",
"Definition:Clopen Set",
"Definition:Open Set/Topology",
"Definition:Disjoint Sets",
"Definition:Open Set/Topo... |
proofwiki-3674 | Partition Space is T3.5 | Let $S$ be a set and let $\PP$ be a partition on $S$.
Let $T = \struct {S, \tau}$ be the partition space whose basis is $\PP$.
Then $T$ is a $T_{3 \frac 1 2}$ space. | Let $F \subseteq S$ be closed.
Denote by $S \setminus F$ the relative complement of $F$ in $S$.
Let $x \in S \setminus F$.
Define a mapping $f: S \to \closedint 0 1$ as:
:<nowiki>$\map f s := \begin{cases}
1 & : \text { if } s \in F \\
0 & : \text { if } s \in S \setminus F
\end{cases}$</nowiki>
Then $f$ is identical... | Let $S$ be a [[Definition:Set|set]] and let $\PP$ be a [[Definition:Partition (Set Theory)|partition]] on $S$.
Let $T = \struct {S, \tau}$ be the [[Definition:Partition Space|partition space]] whose [[Basis for Partition Topology|basis]] is $\PP$.
Then $T$ is a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]]. | Let $F \subseteq S$ be [[Definition:Closed Set (Topology)|closed]].
Denote by $S \setminus F$ the [[Definition:Relative Complement|relative complement]] of $F$ in $S$.
Let $x \in S \setminus F$.
Define a [[Definition:Mapping|mapping]] $f: S \to \closedint 0 1$ as:
:<nowiki>$\map f s := \begin{cases}
1 & : \text {... | Partition Space is T3.5 | https://proofwiki.org/wiki/Partition_Space_is_T3.5 | https://proofwiki.org/wiki/Partition_Space_is_T3.5 | [
"Partition Topologies",
"Examples of T3.5 Spaces"
] | [
"Definition:Set",
"Definition:Set Partition",
"Definition:Partition Topology",
"Basis for Partition Topology",
"Definition:T3.5 Space"
] | [
"Definition:Closed Set/Topology",
"Definition:Relative Complement",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Urysohn Function",
"Definition:T3.5 Space",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Subset of Partition Space is Ope... |
proofwiki-3675 | Odd-Even Topology is Second-Countable | Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$.
Then $T$ is second-countable. | From Basis for Partition Topology, the set:
:$\BB := \set {\set {2 k - 1, 2 k}: k \in \Z, k > 0}$
is a basis for $T$.
There is an obvious one-to-one correspondence $\phi: \Z_{>0} \leftrightarrow \BB$ between $\Z_{>0}$ and $\BB$:
:$\forall x \in \Z_{>0}: \map \phi x = \set {2 x - 1, 2 x}$
But $\Z_{>0} \subseteq \Z$, and... | Let $T = \struct {\Z_{>0}, \tau}$ be a [[Definition:Topological Space|topological space]] where $\tau$ is the [[Definition:Odd-Even Topology|odd-even topology]] on the [[Definition:Strictly Positive Integer|strictly positive integers]] $\Z_{>0}$.
Then $T$ is [[Definition:Second-Countable Space|second-countable]]. | From [[Basis for Partition Topology]], the set:
:$\BB := \set {\set {2 k - 1, 2 k}: k \in \Z, k > 0}$
is a [[Definition:Basis (Topology)|basis]] for $T$.
There is an obvious [[Definition:Bijection|one-to-one correspondence]] $\phi: \Z_{>0} \leftrightarrow \BB$ between $\Z_{>0}$ and $\BB$:
:$\forall x \in \Z_{>0}: \map... | Odd-Even Topology is Second-Countable | https://proofwiki.org/wiki/Odd-Even_Topology_is_Second-Countable | https://proofwiki.org/wiki/Odd-Even_Topology_is_Second-Countable | [
"Odd-Even Topology is Second-Countable",
"Odd-Even Topology",
"Examples of Second-Countable Spaces"
] | [
"Definition:Topological Space",
"Definition:Odd-Even Topology",
"Definition:Strictly Positive/Integer",
"Definition:Second-Countable Space"
] | [
"Basis for Partition Topology",
"Definition:Basis (Topology)",
"Definition:Bijection",
"Integers are Countably Infinite",
"Subset of Countably Infinite Set is Countable",
"Definition:Countable Set",
"Definition:Countable Set",
"Definition:Countable Set",
"Definition:Countable Basis",
"Definition:S... |
proofwiki-3676 | Odd-Even Topology is Weakly Countably Compact | Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$.
Then $T$ is weakly countably compact. | Let $H \subseteq \Z_{>0}$ such that $H$ is infinite.
Let $x \in H$.
By definition, the odd-even topology is a partition topology.
So $U$ is a union of sets of the form $\set {2 k - 1, 2 k}$.
Now if $x \in U$, it will be of the form $2 k - 1$ or $2 k$.
So there will exist $y \in U$ of the form $2 k$ or $2 k - 1$.
So, by... | Let $T = \struct {\Z_{>0}, \tau}$ be a [[Definition:Topological Space|topological space]] where $\tau$ is the [[Definition:Odd-Even Topology|odd-even topology]] on the [[Definition:Strictly Positive Integer|strictly positive integers]] $\Z_{>0}$.
Then $T$ is [[Definition:Weakly Countably Compact Space|weakly countably... | Let $H \subseteq \Z_{>0}$ such that $H$ is [[Definition:Infinite Set|infinite]].
Let $x \in H$.
By definition, the [[Definition:Odd-Even Topology|odd-even topology]] is a [[Definition:Partition Topology|partition topology]].
So $U$ is a [[Definition:Set Union|union]] of sets of the form $\set {2 k - 1, 2 k}$.
Now i... | Odd-Even Topology is Weakly Countably Compact | https://proofwiki.org/wiki/Odd-Even_Topology_is_Weakly_Countably_Compact | https://proofwiki.org/wiki/Odd-Even_Topology_is_Weakly_Countably_Compact | [
"Odd-Even Topology",
"Examples of Weakly Countably Compact Spaces"
] | [
"Definition:Topological Space",
"Definition:Odd-Even Topology",
"Definition:Strictly Positive/Integer",
"Definition:Weakly Countably Compact Space"
] | [
"Definition:Infinite Set",
"Definition:Odd-Even Topology",
"Definition:Partition Topology",
"Definition:Set Union",
"Definition:Limit Point/Topology/Set",
"Definition:Weakly Countably Compact Space"
] |
proofwiki-3677 | Odd-Even Topology is not Countably Compact | Let $T = \struct {\Z_{>0}, \tau}$ be a topological space where $\tau$ is the odd-even topology on the strictly positive integers $\Z_{>0}$.
Then $T$ is not countably compact. | By definition, the odd-even topology is a partition topology.
Let $\PP$ be the partition which is the basis for $T$:
:$\PP = \set {\set {2 k - 1, 2 k}: k \in \Z_{>0} }$
Then $\PP$ is a countable open cover of $S$ which has no finite subcover.
Hence the result.
{{qed}} | Let $T = \struct {\Z_{>0}, \tau}$ be a [[Definition:Topological Space|topological space]] where $\tau$ is the [[Definition:Odd-Even Topology|odd-even topology]] on the [[Definition:Positive Integer|strictly positive integers]] $\Z_{>0}$.
Then $T$ is not [[Definition:Countably Compact Space|countably compact]]. | By definition, the [[Definition:Odd-Even Topology|odd-even topology]] is a [[Definition:Partition Topology|partition topology]].
Let $\PP$ be the [[Definition:Partition (Set Theory)|partition]] which is the [[Basis for Partition Topology|basis for $T$]]:
:$\PP = \set {\set {2 k - 1, 2 k}: k \in \Z_{>0} }$
Then $\PP$ ... | Odd-Even Topology is not Countably Compact | https://proofwiki.org/wiki/Odd-Even_Topology_is_not_Countably_Compact | https://proofwiki.org/wiki/Odd-Even_Topology_is_not_Countably_Compact | [
"Odd-Even Topology",
"Examples of Countably Compact Spaces"
] | [
"Definition:Topological Space",
"Definition:Odd-Even Topology",
"Definition:Positive/Integer",
"Definition:Countably Compact Space"
] | [
"Definition:Odd-Even Topology",
"Definition:Partition Topology",
"Definition:Set Partition",
"Basis for Partition Topology",
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Subcover/Finite"
] |
proofwiki-3678 | Countable Discrete Space is not Weakly Countably Compact | Let $T = \struct {S, \tau}$ be a countable discrete space.
Then $T$ is not weakly countably compact. | Let $A \subseteq S$ be an infinite subset of $S$.
Then as Set in Discrete Topology is Clopen, $A$ is closed in $T$.
From Closed Set Equals its Closure, $A = A^-$ where $A^-$ is the closure of $A$.
From the definition, $x$ is a limit point of $A$ if it belongs to the closure of $A$ but is not an isolated point of $A$.
B... | Let $T = \struct {S, \tau}$ be a [[Definition:Countable Discrete Topology|countable discrete space]].
Then $T$ is not [[Definition:Weakly Countably Compact Space|weakly countably compact]]. | Let $A \subseteq S$ be an [[Definition:Infinite Set|infinite]] subset of $S$.
Then as [[Set in Discrete Topology is Clopen]], $A$ is [[Definition:Closed Set (Topology)|closed]] in $T$.
From [[Closed Set Equals its Closure]], $A = A^-$ where $A^-$ is the [[Definition:Closure (Topology)|closure]] of $A$.
From the defi... | Countable Discrete Space is not Weakly Countably Compact | https://proofwiki.org/wiki/Countable_Discrete_Space_is_not_Weakly_Countably_Compact | https://proofwiki.org/wiki/Countable_Discrete_Space_is_not_Weakly_Countably_Compact | [
"Discrete Topologies",
"Examples of Weakly Countably Compact Spaces"
] | [
"Definition:Discrete Topology/Countable",
"Definition:Weakly Countably Compact Space"
] | [
"Definition:Infinite Set",
"Set in Discrete Topology is Clopen",
"Definition:Closed Set/Topology",
"Set is Closed iff Equals Topological Closure",
"Definition:Closure (Topology)",
"Definition:Limit Point/Topology/Set",
"Definition:Closure (Topology)",
"Definition:Isolated Point (Topology)",
"Topolog... |
proofwiki-3679 | Weak Countable Compactness is not Preserved under Continuous Maps | Let $T_A = \struct {S_A, \tau_A}$ be a topological space which is weakly countably compact.
Let $T_B = \struct {S_B, \tau_B}$ be another topological space.
Let $\phi: T_A \to T_B$ be a continuous mapping.
Then $T_B$ is not necessarily weakly countably compact. | Let $\Z_{>0}$ be the strictly positive integers:
:$\Z_{>0} = \set {1, 2, 3, \ldots}$
Let $T_A = \struct {\Z_{>0}, \tau_A}$ be the odd-even topology.
Let $T_B = \struct {\Z_{>0}, \tau_B}$ be the discrete topology on $\Z_{>0}$.
Let $\phi: T_A \to T_B$ be the mapping:
:$\map \phi {2 k} = k, \map \phi {2 k - 1} = k$
Then:
... | Let $T_A = \struct {S_A, \tau_A}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Weakly Countably Compact Space|weakly countably compact]].
Let $T_B = \struct {S_B, \tau_B}$ be another [[Definition:Topological Space|topological space]].
Let $\phi: T_A \to T_B$ be a [[Definition:Everywh... | Let $\Z_{>0}$ be the [[Definition:Strictly Positive Integer|strictly positive integers]]:
:$\Z_{>0} = \set {1, 2, 3, \ldots}$
Let $T_A = \struct {\Z_{>0}, \tau_A}$ be the [[Definition:Odd-Even Topology|odd-even topology]].
Let $T_B = \struct {\Z_{>0}, \tau_B}$ be the [[Definition:Discrete Topology|discrete topology]]... | Weak Countable Compactness is not Preserved under Continuous Maps | https://proofwiki.org/wiki/Weak_Countable_Compactness_is_not_Preserved_under_Continuous_Maps | https://proofwiki.org/wiki/Weak_Countable_Compactness_is_not_Preserved_under_Continuous_Maps | [
"Weakly Countably Compact Spaces",
"Continuous Mappings (Topology)"
] | [
"Definition:Topological Space",
"Definition:Weakly Countably Compact Space",
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Weakly Countably Compact Space"
] | [
"Definition:Strictly Positive/Integer",
"Definition:Odd-Even Topology",
"Definition:Discrete Topology",
"Definition:Mapping",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Odd-Even Topology is Weakly Countably Compact",
"Countable Discrete Space is not Weakly Countably Compact"
] |
proofwiki-3680 | Deleted Integer Topology is not Countably Compact | Let $S = \R_{\ge 0} \setminus \Z$, and let $\tau$ be the deleted integer topology on $S$.
Then the topological space $T = \struct {S, \tau}$ is not countably compact. | By definition, the deleted integer topology is a partition topology.
Let $\PP$ be the partition which is the basis for $T$, that is:
:$\PP = \set {\openint {n - 1} n: n \in \Z_{> 0} }$
Then $\PP$ is a countable open cover of $S$ which has no finite subcover.
Hence the result.
{{qed}} | Let $S = \R_{\ge 0} \setminus \Z$, and let $\tau$ be the [[Definition:Deleted Integer Topology|deleted integer topology]] on $S$.
Then the [[Definition:Topological Space|topological space]] $T = \struct {S, \tau}$ is not [[Definition:Countably Compact Space|countably compact]]. | By definition, the [[Definition:Deleted Integer Topology|deleted integer topology]] is a [[Definition:Partition Topology|partition topology]].
Let $\PP$ be the [[Definition:Partition (Set Theory)|partition]] which is the [[Basis for Partition Topology|basis for $T$]], that is:
:$\PP = \set {\openint {n - 1} n: n \in ... | Deleted Integer Topology is not Countably Compact | https://proofwiki.org/wiki/Deleted_Integer_Topology_is_not_Countably_Compact | https://proofwiki.org/wiki/Deleted_Integer_Topology_is_not_Countably_Compact | [
"Deleted Integer Topology",
"Examples of Countably Compact Spaces"
] | [
"Definition:Deleted Integer Topology",
"Definition:Topological Space",
"Definition:Countably Compact Space"
] | [
"Definition:Deleted Integer Topology",
"Definition:Partition Topology",
"Definition:Set Partition",
"Basis for Partition Topology",
"Definition:Countable Set",
"Definition:Open Cover",
"Definition:Subcover/Finite"
] |
proofwiki-3681 | Deleted Integer Topology is Second-Countable | Let $S = \R_{\ge 0} \setminus \Z$.
Let $\tau$ be the deleted integer topology on $S$.
Then the topological space $T = \struct {S, \tau}$ is second-countable. | Let $\Z_{>0}$ be understood as the set of strictly positive integers:
:$\Z_{>0} = \set {x \in \Z: x > 0} = \set {1, 2, 3, \ldots}$
From Basis for Partition Topology, the set:
:$\BB = \set {\openint {n - 1} n: n \in Z_{>0} }$
is a basis for $T$.
There is an obvious one-to-one correspondence $\phi: \Z_{> 0} \leftrightarr... | Let $S = \R_{\ge 0} \setminus \Z$.
Let $\tau$ be the [[Definition:Deleted Integer Topology|deleted integer topology]] on $S$.
Then the [[Definition:Topological Space|topological space]] $T = \struct {S, \tau}$ is [[Definition:Second-Countable Space|second-countable]]. | Let $\Z_{>0}$ be understood as the set of [[Definition:Positive Integer|strictly positive integers]]:
:$\Z_{>0} = \set {x \in \Z: x > 0} = \set {1, 2, 3, \ldots}$
From [[Basis for Partition Topology]], the set:
:$\BB = \set {\openint {n - 1} n: n \in Z_{>0} }$
is a [[Definition:Basis (Topology)|basis]] for $T$.
There... | Deleted Integer Topology is Second-Countable | https://proofwiki.org/wiki/Deleted_Integer_Topology_is_Second-Countable | https://proofwiki.org/wiki/Deleted_Integer_Topology_is_Second-Countable | [
"Deleted Integer Topology is Second-Countable",
"Deleted Integer Topology",
"Examples of Second-Countable Spaces"
] | [
"Definition:Deleted Integer Topology",
"Definition:Topological Space",
"Definition:Second-Countable Space"
] | [
"Definition:Positive/Integer",
"Basis for Partition Topology",
"Definition:Basis (Topology)",
"Definition:Bijection",
"Integers are Countably Infinite",
"Subset of Countably Infinite Set is Countable",
"Definition:Countable Set",
"Definition:Countable Set",
"Definition:Countable Basis",
"Definitio... |
proofwiki-3682 | Deleted Integer Topology is Weakly Countably Compact | Let $S = \R_{\ge 0} \setminus \Z$.
Let $\tau$ be the deleted integer topology on $S$.
Then the topological space $T = \struct {S, \tau}$ is weakly countably compact. | Let $A \subseteq S$ such that $A$ is infinite.
Let $x \in A$.
Let $n \in \Z$ such that $n < x < n + 1$.
By the definition of the deleted integer topology, $\openint n {n + 1}$ is open in $T$.
We have that $\openint n {n + 1}$ is infinite.
Take some $y \in \openint n {n + 1}$ such that $x \ne y$.
Now we claim that $y$ i... | Let $S = \R_{\ge 0} \setminus \Z$.
Let $\tau$ be the [[Definition:Deleted Integer Topology|deleted integer topology]] on $S$.
Then the [[Definition:Topological Space|topological space]] $T = \struct {S, \tau}$ is [[Definition:Weakly Countably Compact Space|weakly countably compact]]. | Let $A \subseteq S$ such that $A$ is [[Definition:Infinite Set|infinite]].
Let $x \in A$.
Let $n \in \Z$ such that $n < x < n + 1$.
By the definition of the [[Definition:Deleted Integer Topology|deleted integer topology]], $\openint n {n + 1}$ is [[Definition:Open Set (Topology)|open]] in $T$.
We have that $\openi... | Deleted Integer Topology is Weakly Countably Compact | https://proofwiki.org/wiki/Deleted_Integer_Topology_is_Weakly_Countably_Compact | https://proofwiki.org/wiki/Deleted_Integer_Topology_is_Weakly_Countably_Compact | [
"Deleted Integer Topology",
"Examples of Weakly Countably Compact Spaces"
] | [
"Definition:Deleted Integer Topology",
"Definition:Topological Space",
"Definition:Weakly Countably Compact Space"
] | [
"Definition:Infinite Set",
"Definition:Deleted Integer Topology",
"Definition:Open Set/Topology",
"Definition:Infinite Set",
"Definition:Limit Point/Topology/Set",
"Definition:Open Neighborhood/Point",
"Definition:Open Set/Topology",
"Definition:Set Union",
"Definition:Real Interval/Open",
"Defini... |
proofwiki-3683 | Pseudometric induces Topology | Let $S \ne \O$ be a non-empty set.
Consider a pseudometric space $\struct {S, d}$ where $d: S \times S \to \R_{\ge 0}$ is a pseudometric.
Then $\struct {S, d}$ gives rise to a topological space $\struct {S, \tau_d}$ whose topology $\tau_d$ is '''defined''' (or '''induced''') by $d$. | Let $\tau_d$ be the set of all $X \subseteq S$ which are open in the sense that:
:$\forall y \in X: \exists \epsilon > 0: \map {B_\epsilon} y \subseteq X$
where $\map {B_\epsilon} y$ is the open $\epsilon$-ball of $y$.
Equivalently:
:$\forall x \in X: \exists \epsilon \in \R_{>0}: \forall y \in S: \map d {x, y} < \epsi... | Let $S \ne \O$ be a [[Definition:Non-Empty Set|non-empty set]].
Consider a [[Definition:Pseudometric Space|pseudometric space]] $\struct {S, d}$ where $d: S \times S \to \R_{\ge 0}$ is a [[Definition:Pseudometric|pseudometric]].
Then $\struct {S, d}$ gives rise to a [[Definition:Topological Space|topological space]]... | Let $\tau_d$ be the set of all $X \subseteq S$ which are [[Definition:Open Set (Pseudometric Space)|open]] in the sense that:
:$\forall y \in X: \exists \epsilon > 0: \map {B_\epsilon} y \subseteq X$
where $\map {B_\epsilon} y$ is the [[Definition:Open Ball of Pseudometric Space|open $\epsilon$-ball]] of $y$.
Equival... | Pseudometric induces Topology | https://proofwiki.org/wiki/Pseudometric_induces_Topology | https://proofwiki.org/wiki/Pseudometric_induces_Topology | [
"Topology",
"Pseudometric Spaces"
] | [
"Definition:Non-Empty Set",
"Definition:Pseudometric/Pseudometric Space",
"Definition:Pseudometric",
"Definition:Topological Space",
"Definition:Topology"
] | [
"Definition:Open Set/Pseudometric Space",
"Definition:Open Ball",
"Definition:Topology",
"Axiom:Open Set Axioms",
"Definition:Open Set/Pseudometric Space",
"Definition:Open Set/Pseudometric Space",
"Definition:Open Set/Pseudometric Space",
"Definition:Open Set/Pseudometric Space",
"Definition:Open S... |
proofwiki-3684 | Open Sets in Pseudometric Space | Let $P = \struct {A, d}$ be a pseudometric space.
Then $\O$ and $A$ are both open in $P$. | From the definition of open set:
: $\forall y \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} y \subseteq U$
where $U$ is an open set of $P$.
That is:
:''One cannot get out of $U$ by moving an arbitrarily small distance from any point in $U$.''
Take the case where $U = \O$.
The empty set $\O$ is open by dint of ... | Let $P = \struct {A, d}$ be a [[Definition:Pseudometric Space|pseudometric space]].
Then $\O$ and $A$ are both [[Definition:Open Set (Pseudometric Space)|open]] in $P$. | From the definition of [[Definition:Open Set (Pseudometric Space)|open set]]:
: $\forall y \in U: \exists \epsilon \in \R_{>0}: \map {B_\epsilon} y \subseteq U$
where $U$ is an [[Definition:Open Set (Pseudometric Space)|open set]] of $P$.
That is:
:''One cannot get out of $U$ by moving an arbitrarily small distance fr... | Open Sets in Pseudometric Space | https://proofwiki.org/wiki/Open_Sets_in_Pseudometric_Space | https://proofwiki.org/wiki/Open_Sets_in_Pseudometric_Space | [
"Pseudometric Spaces"
] | [
"Definition:Pseudometric/Pseudometric Space",
"Definition:Open Set/Pseudometric Space"
] | [
"Definition:Open Set/Pseudometric Space",
"Definition:Open Set/Pseudometric Space",
"Definition:Empty Set",
"Definition:Open Set/Pseudometric Space",
"Definition:Vacuous Truth",
"Category:Pseudometric Spaces"
] |
proofwiki-3685 | Partition Space is Pseudometrizable | Let $T = \struct {S, \tau}$ be a partition space.
Then $T$ is pseudometrizable. | Let $\PP$ be the partition which is the basis for $T$.
Let us define $d: S^2 \to \R$ by:
:<nowiki>$\forall x, y \in S: \map d {x, y} = \begin{cases}
0 & : \exists U \in \PP: x, y \in U \\
1 & : \text{otherwise} \end{cases}$</nowiki>
That is, $\map d {x, y} = 0$ when $x$ and $y$ are in the same set in the partition $\PP... | Let $T = \struct {S, \tau}$ be a [[Definition:Partition Space|partition space]].
Then $T$ is [[Definition:Pseudometrizable Topology|pseudometrizable]]. | Let $\PP$ be the [[Definition:Partition (Set Theory)|partition]] which is the [[Basis for Partition Topology|basis for $T$]].
Let us define $d: S^2 \to \R$ by:
:<nowiki>$\forall x, y \in S: \map d {x, y} = \begin{cases}
0 & : \exists U \in \PP: x, y \in U \\
1 & : \text{otherwise} \end{cases}$</nowiki>
That is, $\map... | Partition Space is Pseudometrizable | https://proofwiki.org/wiki/Partition_Space_is_Pseudometrizable | https://proofwiki.org/wiki/Partition_Space_is_Pseudometrizable | [
"Partition Topologies",
"Examples of Pseudometrizable Topologies"
] | [
"Definition:Partition Topology",
"Definition:Pseudometrizable Topology"
] | [
"Definition:Set Partition",
"Basis for Partition Topology",
"Definition:Set Partition",
"Definition:Pseudometric"
] |
proofwiki-3686 | Double Pointed Space is not T0 | Let $T_S = \struct {S, \tau_S}$ be a topological space.
Let $D = \struct {A, \set {\O, A} }$ be the indiscrete space on an arbitrary doubleton $A = \set {a, b}$.
Let $T = \struct {T_S \times D, \tau}$ be the double pointed topological space on $T_S$.
Then $T$ is not a $T_0$ space. | By definition, the double pointed topology $\tau$ on $T_S$ is the product topology on $T_S \times D$.
Let $x \in S$, and consider the point $\tuple {x, a} \in S \times A$.
Then:
{{begin-eqn}}
{{eqn | q = \forall U \in \tau
| l = \tuple {x, a} \in U
| o = \implies
| r = \tuple {x, b} \in U
}}
{{eqn | q... | Let $T_S = \struct {S, \tau_S}$ be a [[Definition:Topological Space|topological space]].
Let $D = \struct {A, \set {\O, A} }$ be the [[Definition:Indiscrete Space|indiscrete space]] on an [[Definition:Arbitrary|arbitrary]] [[Definition:Doubleton|doubleton]] $A = \set {a, b}$.
Let $T = \struct {T_S \times D, \tau}$ be... | By definition, the [[Definition:Double Pointed Topology|double pointed topology]] $\tau$ on $T_S$ is the [[Definition:Product Topology|product topology]] on $T_S \times D$.
Let $x \in S$, and consider the point $\tuple {x, a} \in S \times A$.
Then:
{{begin-eqn}}
{{eqn | q = \forall U \in \tau
| l = \tuple {x,... | Double Pointed Space is not T0/Proof 1 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T0 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T0/Proof_1 | [
"Double Pointed Space is not T0",
"Separation Axioms on Double Pointed Topology",
"Double Pointed Topologies",
"Examples of T0 Spaces"
] | [
"Definition:Topological Space",
"Definition:Indiscrete Topology",
"Definition:Arbitrary",
"Definition:Doubleton",
"Definition:Double Pointed Topology",
"Definition:T0 Space"
] | [
"Definition:Double Pointed Topology",
"Definition:Product Topology",
"Definition:Indiscrete Topology",
"Definition:T0 Space"
] |
proofwiki-3687 | Double Pointed Space is not T0 | Let $T_S = \struct {S, \tau_S}$ be a topological space.
Let $D = \struct {A, \set {\O, A} }$ be the indiscrete space on an arbitrary doubleton $A = \set {a, b}$.
Let $T = \struct {T_S \times D, \tau}$ be the double pointed topological space on $T_S$.
Then $T$ is not a $T_0$ space. | By definition, the double pointed topology $\tau$ on $T_S$ is the product topology on $T_S \times D$.
By definition, $D$ is the indiscrete space on a doubleton.
{{AimForCont}} $T$ is a $T_0$ space.
Then from Product Space is $T_0$ iff Factor Spaces are $T_0$ it follows that $D$ is also a $T_0$ space.
But from Indiscret... | Let $T_S = \struct {S, \tau_S}$ be a [[Definition:Topological Space|topological space]].
Let $D = \struct {A, \set {\O, A} }$ be the [[Definition:Indiscrete Space|indiscrete space]] on an [[Definition:Arbitrary|arbitrary]] [[Definition:Doubleton|doubleton]] $A = \set {a, b}$.
Let $T = \struct {T_S \times D, \tau}$ be... | By definition, the [[Definition:Double Pointed Topology|double pointed topology]] $\tau$ on $T_S$ is the [[Definition:Product Topology|product topology]] on $T_S \times D$.
By definition, $D$ is the [[Definition:Indiscrete Topology|indiscrete space]] on a [[Definition:Doubleton|doubleton]].
{{AimForCont}} $T$ is a [... | Double Pointed Space is not T0/Proof 2 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T0 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T0/Proof_2 | [
"Double Pointed Space is not T0",
"Separation Axioms on Double Pointed Topology",
"Double Pointed Topologies",
"Examples of T0 Spaces"
] | [
"Definition:Topological Space",
"Definition:Indiscrete Topology",
"Definition:Arbitrary",
"Definition:Doubleton",
"Definition:Double Pointed Topology",
"Definition:T0 Space"
] | [
"Definition:Double Pointed Topology",
"Definition:Product Topology",
"Definition:Indiscrete Topology",
"Definition:Doubleton",
"Definition:T0 Space",
"Product Space is T0 iff Factor Spaces are T0",
"Definition:T0 Space",
"Indiscrete Non-Singleton Space is not T0",
"Definition:T0 Space",
"Proof by ... |
proofwiki-3688 | Double Pointed Discrete Real Number Space is Weakly Countably Compact | Let $T_\R = \struct {\R, \tau_\R}$ be the (uncountable) discrete space on the set of real numbers.
Let $T_D = \struct {D, \tau_D}$ be the indiscrete topology on the doubleton $D = \set {a, b}$.
Let $T = T_\R \times T_D$ be the double pointed (uncountable) discrete space which is the product space of $T_\R$ and $T_D$.
T... | We have that $T$ is a partition topology, whose basis $\PP$ is defined as:
:$\PP = \set {\set {\tuple {s, a}, \tuple {s, b} }: s \in \R}$
Let $A \subseteq \R \times D$ such that $A$ is infinite.
Let $x \in A$.
Let $U$ be the union of sets of the form $\set {\tuple {s, a}, \tuple {s, b} }$, and hence open in $T$.
Now if... | Let $T_\R = \struct {\R, \tau_\R}$ be the [[Definition:Uncountable Discrete Topology|(uncountable) discrete space]] on the [[Definition:Real Numbers|set of real numbers]].
Let $T_D = \struct {D, \tau_D}$ be the [[Definition:Indiscrete Topology|indiscrete topology]] on the [[Definition:Doubleton|doubleton]] $D = \set {... | We have that $T$ is a [[Definition:Partition Topology|partition topology]], whose [[Basis for Partition Topology|basis]] $\PP$ is defined as:
:$\PP = \set {\set {\tuple {s, a}, \tuple {s, b} }: s \in \R}$
Let $A \subseteq \R \times D$ such that $A$ is [[Definition:Infinite Set|infinite]].
Let $x \in A$.
Let $U$ be t... | Double Pointed Discrete Real Number Space is Weakly Countably Compact | https://proofwiki.org/wiki/Double_Pointed_Discrete_Real_Number_Space_is_Weakly_Countably_Compact | https://proofwiki.org/wiki/Double_Pointed_Discrete_Real_Number_Space_is_Weakly_Countably_Compact | [
"Double Pointed Topologies",
"Discrete Topologies",
"Partition Topologies",
"Examples of Weakly Countably Compact Spaces"
] | [
"Definition:Discrete Topology/Uncountable",
"Definition:Real Number",
"Definition:Indiscrete Topology",
"Definition:Doubleton",
"Definition:Double Pointed Topology",
"Definition:Discrete Topology/Uncountable",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Weakly Countably Compac... | [
"Definition:Partition Topology",
"Basis for Partition Topology",
"Definition:Infinite Set",
"Definition:Set Union",
"Definition:Set",
"Definition:Open Set/Topology",
"Definition:Element",
"Definition:Element",
"Definition:Limit Point/Topology/Set",
"Definition:Weakly Countably Compact Space"
] |
proofwiki-3689 | Double Pointed Discrete Real Number Space is not Lindelöf | Let $T_\R = \struct {\R, \tau_\R}$ be the (uncountable) discrete space on the set of real numbers.
Let $T_D = \struct {D, \tau_D}$ be the indiscrete topology on the doubleton $D = \set {a, b}$.
Let $T = T_\R \times T_D$ be the double pointed (uncountable) discrete space which is the product space of $T_\R$ and $T_D$.
T... | We have that $T$ is a partition topology, whose basis $\PP$ is defined as:
:$\PP = \set {\set {\paren {s, a}, \paren {s, b} }: s \in \R}$
We have that $\PP$ is an open cover of $T$.
But $\PP$ has no countable subcover.
Hence the result, by definition of Lindelöf space.
{{qed}} | Let $T_\R = \struct {\R, \tau_\R}$ be the [[Definition:Uncountable Discrete Topology|(uncountable) discrete space]] on the [[Definition:Real Numbers|set of real numbers]].
Let $T_D = \struct {D, \tau_D}$ be the [[Definition:Indiscrete Topology|indiscrete topology]] on the [[Definition:Doubleton|doubleton]] $D = \set {... | We have that $T$ is a [[Definition:Partition Topology|partition topology]], whose [[Basis for Partition Topology|basis]] $\PP$ is defined as:
:$\PP = \set {\set {\paren {s, a}, \paren {s, b} }: s \in \R}$
We have that $\PP$ is an [[Definition:Open Cover|open cover]] of $T$.
But $\PP$ has no [[Definition:Countable Sub... | Double Pointed Discrete Real Number Space is not Lindelöf | https://proofwiki.org/wiki/Double_Pointed_Discrete_Real_Number_Space_is_not_Lindelöf | https://proofwiki.org/wiki/Double_Pointed_Discrete_Real_Number_Space_is_not_Lindelöf | [
"Double Pointed Topologies",
"Discrete Topologies",
"Examples of Lindelöf Spaces",
"Partition Topologies"
] | [
"Definition:Discrete Topology/Uncountable",
"Definition:Real Number",
"Definition:Indiscrete Topology",
"Definition:Doubleton",
"Definition:Double Pointed Topology",
"Definition:Discrete Topology/Uncountable",
"Definition:Product Space (Topology)/Two Factor Spaces",
"Definition:Lindelöf Space"
] | [
"Definition:Partition Topology",
"Basis for Partition Topology",
"Definition:Open Cover",
"Definition:Subcover/Countable",
"Definition:Lindelöf Space"
] |
proofwiki-3690 | Particular Point Topology is Topology | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $\tau_p$ is a topology on $S$, and $T$ is a topological space. | We have by definition that $\O \in \tau_p$, and as $p \in S$ we have that $S \in \tau_p$.
Now let $U_1, U_2 \in \tau_p$.
By definition $p \in U_1$ and $p \in U_2$, and so $p \in U_1 \cap U_2$ by definition of set intersection.
So $U_1 \cap U_2 \in \tau_p$.
Now let $\UU \subseteq \tau_p$.
We have that $\forall U \in \UU... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Space|particular point space]].
Then $\tau_p$ is a [[Definition:Topology|topology]] on $S$, and $T$ is a [[Definition:Topological Space|topological space]]. | We have [[Definition:Particular Point Topology|by definition]] that $\O \in \tau_p$, and as $p \in S$ we have that $S \in \tau_p$.
Now let $U_1, U_2 \in \tau_p$.
By definition $p \in U_1$ and $p \in U_2$, and so $p \in U_1 \cap U_2$ by definition of [[Definition:Set Intersection|set intersection]].
So $U_1 \cap U_2... | Particular Point Topology is Topology | https://proofwiki.org/wiki/Particular_Point_Topology_is_Topology | https://proofwiki.org/wiki/Particular_Point_Topology_is_Topology | [
"Particular Point Topologies"
] | [
"Definition:Particular Point Topology",
"Definition:Topology",
"Definition:Topological Space"
] | [
"Definition:Particular Point Topology",
"Definition:Set Intersection",
"Set is Subset of Union",
"Definition:Topology"
] |
proofwiki-3691 | Accumulation Points of Sequence of Distinct Terms in Infinite Particular Point Space | Let $T = \struct {S, \tau_p}$ be an infinite particular point space.
Let $U \in \tau_p$ be a countably infinite open set of $T$.
Let the elements of $U$ be arranged into a sequence $\sequence {a_i}$ of distinct terms in $T$.
Then while every element $x$ of $U$ such that $x \ne p$ is a limit point of $U$, there exists n... | We note from Limit Points in Particular Point Space that every $x \in U$ such that $x \ne p$ is a limit point of $U$.
Let $\sequence {a_i}$ be the (infinite) sequence in $T$ whose terms are the elements of $U$.
{{AimForCont}} $\beta \in S$ is an accumulation point of $\sequence {a_i}$.
Then from Accumulation Points for... | Let $T = \struct {S, \tau_p}$ be an [[Definition:Infinite Particular Point Topology|infinite particular point space]].
Let $U \in \tau_p$ be a [[Definition:Countably Infinite Set|countably infinite]] [[Definition:Open Set (Topology)|open set]] of $T$.
Let the [[Definition:Element|elements]] of $U$ be arranged into a ... | We note from [[Limit Points in Particular Point Space]] that every $x \in U$ such that $x \ne p$ is a [[Definition:Limit Point of Set|limit point]] of $U$.
Let $\sequence {a_i}$ be the [[Definition:Infinite Sequence|(infinite) sequence]] in $T$ whose [[Definition:Term of Sequence|terms]] are the [[Definition:Element|... | Accumulation Points of Sequence of Distinct Terms in Infinite Particular Point Space | https://proofwiki.org/wiki/Accumulation_Points_of_Sequence_of_Distinct_Terms_in_Infinite_Particular_Point_Space | https://proofwiki.org/wiki/Accumulation_Points_of_Sequence_of_Distinct_Terms_in_Infinite_Particular_Point_Space | [
"Infinite Particular Point Topologies",
"Examples of Accumulation Points"
] | [
"Definition:Particular Point Topology/Infinite",
"Definition:Countably Infinite/Set",
"Definition:Open Set/Topology",
"Definition:Element",
"Definition:Sequence of Distinct Terms",
"Definition:Element",
"Definition:Limit Point/Topology/Set",
"Definition:Accumulation Point/Sequence"
] | [
"Limit Points in Particular Point Space",
"Definition:Limit Point/Topology/Set",
"Definition:Sequence/Infinite Sequence",
"Definition:Term of Sequence",
"Definition:Element",
"Definition:Accumulation Point/Sequence",
"Accumulation Points for Sequence in Particular Point Space",
"Definition:Infinite Se... |
proofwiki-3692 | Closure of Open Set of Particular Point Space | Let $T = \struct {S, \tau_p}$ be a particular point space.
Let $U \in \tau_p$ be open in $T$ such that $U \ne \O$.
Then:
:$U^- = S$
where $U^-$ denotes the closure of $U$. | Follows directly from:
:Particular Point Topology is Closed Extension Topology of Discrete Topology
:Closure of Open Set of Closed Extension Space
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Let $U \in \tau_p$ be [[Definition:Open Set (Topology)|open]] in $T$ such that $U \ne \O$.
Then:
:$U^- = S$
where $U^-$ denotes the [[Definition:Closure (Topology)|closure]] of $U$. | Follows directly from:
:[[Particular Point Topology is Closed Extension Topology of Discrete Topology]]
:[[Closure of Open Set of Closed Extension Space]]
{{qed}} | Closure of Open Set of Particular Point Space | https://proofwiki.org/wiki/Closure_of_Open_Set_of_Particular_Point_Space | https://proofwiki.org/wiki/Closure_of_Open_Set_of_Particular_Point_Space | [
"Particular Point Topologies",
"Examples of Set Closures",
"Open Sets"
] | [
"Definition:Particular Point Topology",
"Definition:Open Set/Topology",
"Definition:Closure (Topology)"
] | [
"Particular Point Topology is Closed Extension Topology of Discrete Topology",
"Closure of Open Set of Closed Extension Space"
] |
proofwiki-3693 | Interior of Closed Set of Particular Point Space | Let $T = \struct {S, \tau_p}$ be a particular point space.
Let $V \subseteq S$ be closed in $T$ such that $V \ne S$.
Then:
:$V^\circ = \O$
where $V^\circ$ denotes the interior of $V$. | By definition:
:$\forall U \in \tau_p, U \ne \O: p \in U$
Thus if $V$ is closed in $T$:
:$\exists U \subseteq T: V = \relcomp S U$
So $p \notin V$.
Hence no open set of $T$ can be a subset of $V$ unless it is $\O$.
Hence the result, by definition of interior.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]].
Let $V \subseteq S$ be [[Definition:Closed Set (Topology)|closed]] in $T$ such that $V \ne S$.
Then:
:$V^\circ = \O$
where $V^\circ$ denotes the [[Definition:Interior (Topology)|interior]] of $V$. | By definition:
:$\forall U \in \tau_p, U \ne \O: p \in U$
Thus if $V$ is [[Definition:Closed Set (Topology)|closed]] in $T$:
:$\exists U \subseteq T: V = \relcomp S U$
So $p \notin V$.
Hence no [[Definition:Open Set (Topology)|open set]] of $T$ can be a [[Definition:Subset|subset]] of $V$ unless it is $\O$.
Hence... | Interior of Closed Set of Particular Point Space | https://proofwiki.org/wiki/Interior_of_Closed_Set_of_Particular_Point_Space | https://proofwiki.org/wiki/Interior_of_Closed_Set_of_Particular_Point_Space | [
"Particular Point Topologies",
"Examples of Set Interiors",
"Closed Sets"
] | [
"Definition:Particular Point Topology",
"Definition:Closed Set/Topology",
"Definition:Interior (Topology)"
] | [
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:Subset",
"Definition:Interior (Topology)"
] |
proofwiki-3694 | Point in Open Set of Particular Point Space is not Omega-Accumulation Point | Let $T = \struct {S, \tau_p}$ be a particular point space.
Let $x \in S$ such that $x \ne p$.
Let $H \subseteq S$ such that $p \in H$.
Then $x$ is not an $\omega$-accumulation point of $H$. | Let $x \in S, x \ne p$.
By Limit Points in Particular Point Space, $x$ is a limit point of $H$.
Consider the set $U = \set {x, p} \subseteq S$.
By definition of the particular point topology, $U$ is open in $T$.
But as $U$ contains only $x$ and $p$, it is clear that $U$ does not contain an infinite number of points of ... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Space|particular point space]].
Let $x \in S$ such that $x \ne p$.
Let $H \subseteq S$ such that $p \in H$.
Then $x$ is not an [[Definition:Omega-Accumulation Point|$\omega$-accumulation point]] of $H$. | Let $x \in S, x \ne p$.
By [[Limit Points in Particular Point Space]], $x$ is a [[Definition:Limit Point of Set|limit point]] of $H$.
Consider the set $U = \set {x, p} \subseteq S$.
By definition of the [[Definition:Particular Point Topology|particular point topology]], $U$ is [[Definition:Open Set (Topology)|open]]... | Point in Open Set of Particular Point Space is not Omega-Accumulation Point | https://proofwiki.org/wiki/Point_in_Open_Set_of_Particular_Point_Space_is_not_Omega-Accumulation_Point | https://proofwiki.org/wiki/Point_in_Open_Set_of_Particular_Point_Space_is_not_Omega-Accumulation_Point | [
"Particular Point Topologies",
"Examples of Omega-Accumulation Points"
] | [
"Definition:Particular Point Topology",
"Definition:Omega-Accumulation Point"
] | [
"Limit Points in Particular Point Space",
"Definition:Limit Point/Topology/Set",
"Definition:Particular Point Topology",
"Definition:Open Set/Topology",
"Definition:Infinite",
"Definition:Element",
"Definition:Omega-Accumulation Point"
] |
proofwiki-3695 | Particular Point Space is T0 | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is a $T_0$ space. | Let $T$ be a trivial space, so that $S = \set p$.
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 $\set p \subseteq T$ is open in $T$ such that $p \in \set p$ but $x \notin \set p$.
Finally, suppose that $x, y ... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Space|particular point space]].
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | Let $T$ be a [[Definition:Trivial Topological Space|trivial space]], so that $S = \set p$.
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 $\exists ... | Particular Point Space is T0/Proof 1 | https://proofwiki.org/wiki/Particular_Point_Space_is_T0 | https://proofwiki.org/wiki/Particular_Point_Space_is_T0/Proof_1 | [
"Particular Point Space is T0",
"Particular Point Topologies",
"Examples of T0 Spaces"
] | [
"Definition:Particular 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-3696 | Particular Point Space is T0 | Let $T = \struct {S, \tau_p}$ be a particular point space.
Then $T$ is a $T_0$ space. | We have:
:Particular Point Topology is Closed Extension Topology of Discrete Topology
:Discrete Space satisfies all Separation Properties (including being a $T_0$ space)
Then by Condition for Closed Extension Space to be $T_0$ Space, as a discrete space is $T_0$ then so is its closed extension.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Space|particular point space]].
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | We have:
:[[Particular Point Topology is Closed Extension Topology of Discrete Topology]]
:[[Discrete Space satisfies all Separation Properties]] (including being a [[Definition:T0 Space|$T_0$ space]])
Then by [[Condition for Closed Extension Space to be T0 Space|Condition for Closed Extension Space to be $T_0$ Spac... | Particular Point Space is T0/Proof 2 | https://proofwiki.org/wiki/Particular_Point_Space_is_T0 | https://proofwiki.org/wiki/Particular_Point_Space_is_T0/Proof_2 | [
"Particular Point Space is T0",
"Particular Point Topologies",
"Examples of T0 Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:T0 Space"
] | [
"Particular Point Topology is Closed Extension Topology of Discrete Topology",
"Discrete Space satisfies all Separation Properties",
"Definition:T0 Space",
"Condition for Closed Extension Space to be T0 Space",
"Definition:Discrete Topology",
"Definition:Closed Extension Topology"
] |
proofwiki-3697 | Non-Trivial Particular Point Space is not T1 | Let $T = \struct {S, \tau_p}$ be a particular point space such that $S$ is not a singleton.
Then $T$ is not a $T_1$ space. | Follows directly from:
:Particular Point Topology is Closed Extension Topology of Discrete Topology
:Closed Extension Topology is not $T_1$.
{{qed}} | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Space|particular point space]] such that $S$ is not a [[Definition:Singleton|singleton]].
Then $T$ is not a [[Definition:T1 Space|$T_1$ space]]. | Follows directly from:
:[[Particular Point Topology is Closed Extension Topology of Discrete Topology]]
:[[Closed Extension Topology is not T1|Closed Extension Topology is not $T_1$]].
{{qed}} | Non-Trivial Particular Point Space is not T1 | https://proofwiki.org/wiki/Non-Trivial_Particular_Point_Space_is_not_T1 | https://proofwiki.org/wiki/Non-Trivial_Particular_Point_Space_is_not_T1 | [
"Particular Point Topologies",
"Examples of T1 Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Singleton",
"Definition:T1 Space"
] | [
"Particular Point Topology is Closed Extension Topology of Discrete Topology",
"Closed Extension Topology is not T1"
] |
proofwiki-3698 | Particular Point Topology with Three Points is not T4 | Let $T = \struct {S, \tau_p}$ be a particular point space such that $S$ is not a singleton or a doubleton.
That is, such that $S$ has more than two distinct elements.
Then $T$ is not a $T_4$ space. | We have that there are at least three elements of $S$.
So, consider $x, y, p \in S: x \ne y, x \ne p, y \ne p$.
Then $X = \set x, Y = \set y$ are closed in $T$ and $X \cap Y = \O$.
Suppose $U, V \in \tau_p$ are open sets in $T$ such that $X \subseteq U, Y \subseteq V$.
But as $p \in U, p \in V$ we have that $U \cap V \... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]] such that $S$ is not a [[Definition:Singleton|singleton]] or a [[Definition:Doubleton|doubleton]].
That is, such that $S$ has more than two [[Definition:Distinct Elements|distinct elements]].
Then $T$ is not a [[Defini... | We have that there are at least three [[Definition:Element|elements]] of $S$.
So, consider $x, y, p \in S: x \ne y, x \ne p, y \ne p$.
Then $X = \set x, Y = \set y$ are [[Definition:Closed Set (Topology)|closed]] in $T$ and $X \cap Y = \O$.
Suppose $U, V \in \tau_p$ are [[Definition:Open Set (Topology)|open sets]] i... | Particular Point Topology with Three Points is not T4 | https://proofwiki.org/wiki/Particular_Point_Topology_with_Three_Points_is_not_T4 | https://proofwiki.org/wiki/Particular_Point_Topology_with_Three_Points_is_not_T4 | [
"Particular Point Topologies",
"Examples of T4 Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Singleton",
"Definition:Doubleton",
"Definition:Distinct/Plural",
"Definition:T4 Space"
] | [
"Definition:Element",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:T4 Space"
] |
proofwiki-3699 | Non-Trivial Particular Point Topology is not T3 | Let $T = \struct {S, \tau_p}$ be a particular point space such that $S$ is not a singleton.
Then $T$ is not a $T_3$ space. | We have that there are at least two distinct elements of $S$.
So, consider $x, p \in S: x \ne p$.
Then $X = \set x$ is closed in $T$ and $p \notin X$.
Suppose $U \in \tau_p$ is an open set in $T$ such that $X \subseteq U$.
We have that $\set p \in \tau_p$ such that $p \in \set p$.
But as $p \in U, p \in \set p$ we have... | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]] such that $S$ is not a [[Definition:Singleton|singleton]].
Then $T$ is not a [[Definition:T3 Space|$T_3$ space]]. | We have that there are at least two [[Definition:Distinct Elements|distinct elements]] of $S$.
So, consider $x, p \in S: x \ne p$.
Then $X = \set x$ is [[Definition:Closed Set (Topology)|closed]] in $T$ and $p \notin X$.
Suppose $U \in \tau_p$ is an [[Definition:Open Set (Topology)|open set]] in $T$ such that $X \su... | Non-Trivial Particular Point Topology is not T3 | https://proofwiki.org/wiki/Non-Trivial_Particular_Point_Topology_is_not_T3 | https://proofwiki.org/wiki/Non-Trivial_Particular_Point_Topology_is_not_T3 | [
"Particular Point Topologies",
"Examples of T3 Spaces"
] | [
"Definition:Particular Point Topology",
"Definition:Singleton",
"Definition:T3 Space"
] | [
"Definition:Distinct/Plural",
"Definition:Closed Set/Topology",
"Definition:Open Set/Topology",
"Definition:T3 Space"
] |
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