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-23300 | Real Number Line with Euclidean Topology is Fully Normal | :$\struct {\R, \tau_d}$ is a fully normal space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is Fully Normal.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Fully Normal Spaces
dv... | :$\struct {\R, \tau_d}$ is a [[Definition:Fully Normal Space|fully normal space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is Fully Normal]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclidean Topology... | Real Number Line with Euclidean Topology is Fully Normal | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Fully_Normal | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Fully_Normal | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of Fully Normal Spaces"
] | [
"Definition:Fully Normal Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is Fully Normal",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of Fully Normal Spaces"
] |
proofwiki-23301 | Real Number Line with Euclidean Topology is Perfectly Normal | :$\struct {\R, \tau_d}$ is a perfectly normal space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is Perfectly Normal.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Perfectly Normal S... | :$\struct {\R, \tau_d}$ is a [[Definition:Perfectly Normal Space|perfectly normal space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is Perfectly Normal]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclidean Topo... | Real Number Line with Euclidean Topology is Perfectly Normal | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Perfectly_Normal | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Perfectly_Normal | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of Perfectly Normal Spaces"
] | [
"Definition:Perfectly Normal Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is Perfectly Normal",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of Perfectly Normal Spaces"
] |
proofwiki-23302 | Real Number Line with Euclidean Topology is Normal | :$\struct {\R, \tau_d}$ is a normal space | From Real Number Line is Metric Space we have that $\struct {\R, \tau_d}$ is an example of a metric space.
The result follows from Metric Space is Normal.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Normal Spaces
seofn6xolwmzqq... | :$\struct {\R, \tau_d}$ is a [[Definition:Normal Space|normal space]] | From [[Real Number Line is Metric Space]] we have that $\struct {\R, \tau_d}$ is an example of a [[Definition:Metric Space|metric space]].
The result follows from [[Metric Space is Normal]].
{{qed}}
[[Category:Real Number Line satisfies all Separation Axioms]]
[[Category:Real Number Line with Euclidean Topology]]
[[C... | Real Number Line with Euclidean Topology is Normal | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Normal | https://proofwiki.org/wiki/Real_Number_Line_with_Euclidean_Topology_is_Normal | [
"Real Number Line satisfies all Separation Axioms",
"Real Number Line with Euclidean Topology",
"Examples of Normal Spaces"
] | [
"Definition:Normal Space"
] | [
"Real Number Line is Metric Space",
"Definition:Metric Space",
"Metric Space is Normal",
"Category:Real Number Line satisfies all Separation Axioms",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of Normal Spaces"
] |
proofwiki-23303 | Indiscrete Extension of Reals is T1 | Let $T$ be an indiscrete extension of $\R$.
Then $T$ is a $T_1$ space. | We have by definition that an indiscrete extension of $\R$ is an expansion of $\struct {\R, \tau_d}$.
From Real Number Line with Euclidean Topology is $T_1$, $\struct {\R, \tau_d}$ is a $T_1$ space.
The result follows from $T_1$ Property is Preserved under Expansion.
{{qed}}
Category:Indiscrete Extensions of Reals
Cate... | Let $T$ be an [[Definition:Indiscrete Extension of Reals|indiscrete extension of $\R$]].
Then $T$ is a [[Definition:T1 Space|$T_1$ space]]. | We have by definition that an [[Definition:Indiscrete Extension of Reals|indiscrete extension of $\R$]] is an [[Definition:Expansion of Topology|expansion]] of $\struct {\R, \tau_d}$.
From [[Real Number Line with Euclidean Topology is T1|Real Number Line with Euclidean Topology is $T_1$]], $\struct {\R, \tau_d}$ is a ... | Indiscrete Extension of Reals is T1 | https://proofwiki.org/wiki/Indiscrete_Extension_of_Reals_is_T1 | https://proofwiki.org/wiki/Indiscrete_Extension_of_Reals_is_T1 | [
"Indiscrete Extensions of Reals",
"Examples of T1 Spaces"
] | [
"Definition:Indiscrete Extension of Reals",
"Definition:T1 Space"
] | [
"Definition:Indiscrete Extension of Reals",
"Definition:Expansion of Topology",
"Real Number Line with Euclidean Topology is T1",
"Definition:T1 Space",
"T1 Property is Preserved under Expansion",
"Category:Indiscrete Extensions of Reals",
"Category:Examples of T1 Spaces"
] |
proofwiki-23304 | Indiscrete Extension of Reals is T0 | Let $T$ be an indiscrete extension of $\R$.
Then $T$ is a $T_0$ space. | We have by definition that an indiscrete extension of $\R$ is an expansion of $\struct {\R, \tau_d}$.
From Real Number Line with Euclidean Topology is $T_0$, $\struct {\R, \tau_d}$ is a $T_0$ space.
The result follows from $T_0$ Property is Preserved under Expansion.
{{qed}}
Category:Indiscrete Extensions of Reals
Cate... | Let $T$ be an [[Definition:Indiscrete Extension of Reals|indiscrete extension of $\R$]].
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | We have by definition that an [[Definition:Indiscrete Extension of Reals|indiscrete extension of $\R$]] is an [[Definition:Expansion of Topology|expansion]] of $\struct {\R, \tau_d}$.
From [[Real Number Line with Euclidean Topology is T0|Real Number Line with Euclidean Topology is $T_0$]], $\struct {\R, \tau_d}$ is a ... | Indiscrete Extension of Reals is T0 | https://proofwiki.org/wiki/Indiscrete_Extension_of_Reals_is_T0 | https://proofwiki.org/wiki/Indiscrete_Extension_of_Reals_is_T0 | [
"Indiscrete Extensions of Reals",
"Examples of T0 Spaces"
] | [
"Definition:Indiscrete Extension of Reals",
"Definition:T0 Space"
] | [
"Definition:Indiscrete Extension of Reals",
"Definition:Expansion of Topology",
"Real Number Line with Euclidean Topology is T0",
"Definition:T0 Space",
"T0 Property is Preserved under Expansion",
"Category:Indiscrete Extensions of Reals",
"Category:Examples of T0 Spaces"
] |
proofwiki-23305 | Regular Property is Hereditary | :The property of being a regular space is hereditary. | Let $T = \struct {S, \tau}$ be a regular space.
{{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
From:
:$T_0$ Property is Hereditary
:$T_3$ Property is Hereditary
follows the result.
{{qed}} | :The property of being a [[Definition:Regular Space|regular space]] is [[Definition:Hereditary Property (Topology)|hereditary]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Regular Space|regular space]].
{{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
From:
:[[T0 Property is Hereditary|$T_0$ Property is Hereditary]]
:[[T3 Property is Hereditary|$T_3$ Property is Hereditary]]
follows the result.
{{qed}} | Regular Property is Hereditary | https://proofwiki.org/wiki/Regular_Property_is_Hereditary | https://proofwiki.org/wiki/Regular_Property_is_Hereditary | [
"Regular Spaces",
"Separation Properties Preserved in Subspace",
"Examples of Hereditary Properties"
] | [
"Definition:Regular Space",
"Definition:Hereditary Property (Topology)"
] | [
"Definition:Regular Space",
"T0 Property is Hereditary",
"T3 Property is Hereditary"
] |
proofwiki-23306 | Completely Regular Property is Hereditary | :The property of being a completely regular space is hereditary. | Let $T = \struct {S, \tau}$ be a completely regular space.
{{Recall|Completely Regular Space|completely regular space|index = 1}}
{{:Definition:Completely Regular Space/Definition 1}}
From:
:$T_0$ Property is Hereditary
:$T_{3 \frac 1 2}$ Property is Hereditary
follows the result.
{{qed}} | :The property of being a [[Definition:Completely Regular Space|completely regular space]] is [[Definition:Hereditary Property (Topology)|hereditary]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Completely Regular Space|completely regular space]].
{{Recall|Completely Regular Space|completely regular space|index = 1}}
{{:Definition:Completely Regular Space/Definition 1}}
From:
:[[T0 Property is Hereditary|$T_0$ Property is Hereditary]]
:[[T3.5 Property is Heredita... | Completely Regular Property is Hereditary | https://proofwiki.org/wiki/Completely_Regular_Property_is_Hereditary | https://proofwiki.org/wiki/Completely_Regular_Property_is_Hereditary | [
"Completely Regular Spaces",
"Separation Properties Preserved in Subspace",
"Examples of Hereditary Properties"
] | [
"Definition:Completely Regular Space",
"Definition:Hereditary Property (Topology)"
] | [
"Definition:Completely Regular Space",
"T0 Property is Hereditary",
"T3.5 Property is Hereditary"
] |
proofwiki-23307 | Completely Normal Property is Hereditary | :The property of being a completely normal space is hereditary. | Let $T = \struct {S, \tau}$ be a completely normal space.
{{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
From:
{{begin-itemize}}
{{item||$T_1$ Property is Hereditary}}
{{item||$T_5$ Property is Hereditary}}
{{end-itemize}}
follows the result.
{{... | :The property of being a [[Definition:Completely Normal Space|completely normal space]] is [[Definition:Hereditary Property (Topology)|hereditary]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Completely Normal Space|completely normal space]].
{{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
From:
{{begin-itemize}}
{{item||[[T1 Property is Hereditary|$T_1$ Property is Hereditary]]}}
{{ite... | Completely Normal Property is Hereditary | https://proofwiki.org/wiki/Completely_Normal_Property_is_Hereditary | https://proofwiki.org/wiki/Completely_Normal_Property_is_Hereditary | [
"Completely Normal Spaces",
"Separation Properties Preserved in Subspace",
"Examples of Hereditary Properties"
] | [
"Definition:Completely Normal Space",
"Definition:Hereditary Property (Topology)"
] | [
"Definition:Completely Normal Space",
"T1 Property is Hereditary",
"T5 Property is Hereditary"
] |
proofwiki-23308 | Normal Property is not Hereditary | Let $T = \struct {S, \tau}$ be a topological space which is a normal space.
Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a subspace of $T$.
Then it does not necessarily follow that $T_H$ is a normal space. | Let $T$ be the Tychonoff plank.
Let $T'$ be the deleted Tychonoff plank.
By definition, $T'$ is a subspace of $T$.
From Tychonoff Plank is Normal, $T$ is a normal space.
From Deleted Tychonoff Plank is Not Normal, $T'$ is not a normal space.
Thus it is seen that the property of being a normal space is not inherited by ... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is a [[Definition:Normal Space|normal space]].
Let $T_H = \struct {H, \tau_H}$, where $\O \subset H \subseteq S$, be a [[Definition:Topological Subspace|subspace]] of $T$.
Then it does not necessarily follow that $T_H$ is a [[D... | Let $T$ be the [[Definition:Tychonoff Plank|Tychonoff plank]].
Let $T'$ be the [[Definition:Deleted Tychonoff Plank|deleted Tychonoff plank]].
By definition, $T'$ is a [[Definition:Topological Subspace|subspace]] of $T$.
From [[Tychonoff Plank is Normal]], $T$ is a [[Definition:Normal Space|normal space]].
From [[... | Normal Property is not Hereditary | https://proofwiki.org/wiki/Normal_Property_is_not_Hereditary | https://proofwiki.org/wiki/Normal_Property_is_not_Hereditary | [
"Normal Spaces",
"Examples of Hereditary Properties",
"Separation Properties Preserved in Subspace"
] | [
"Definition:Topological Space",
"Definition:Normal Space",
"Definition:Topological Subspace",
"Definition:Normal Space"
] | [
"Definition:Tychonoff Plank",
"Definition:Tychonoff Plank/Deleted",
"Definition:Topological Subspace",
"Tychonoff Plank is Normal",
"Definition:Normal Space",
"Deleted Tychonoff Plank is Not Normal",
"Definition:Normal Space",
"Definition:Normal Space",
"Definition:Topological Subspace"
] |
proofwiki-23309 | T2.5 Space is not necessarily Urysohn | A $T_{2 \frac 1 2}$ topological space is not necessarily an Urysohn space. | Let $T$ be an Arens square.
From Arens Square is $T_{2 \frac 1 2}$, $T$ is a $T_{2 \frac 1 2}$ space.
From Arens Square is not Urysohn, $T$ is not a Urysohn space.
Hence the result.
{{qed}} | A [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ topological space]] is not necessarily an [[Definition:Urysohn Space|Urysohn space]]. | Let $T$ be an [[Definition:Arens Square|Arens square]].
From [[Arens Square is T2.5|Arens Square is $T_{2 \frac 1 2}$]], $T$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]].
From [[Arens Square is not Urysohn]], $T$ is not a [[Definition:Urysohn Space|Urysohn space]].
Hence the result.
{{qed}} | T2.5 Space is not necessarily Urysohn | https://proofwiki.org/wiki/T2.5_Space_is_not_necessarily_Urysohn | https://proofwiki.org/wiki/T2.5_Space_is_not_necessarily_Urysohn | [
"T2.5 Spaces",
"Urysohn Spaces"
] | [
"Definition:T2.5 Space",
"Definition:Urysohn Space"
] | [
"Definition:Arens Square",
"Arens Square is T2.5",
"Definition:T2.5 Space",
"Arens Square is not Urysohn",
"Definition:Urysohn Space"
] |
proofwiki-23310 | Double Pointed Finite Complement Space is not T0 | :$T \times D$ is not a $T_0$ space | From Double Pointed Space is not $T_0$:
:$T \times D$ is not a $T_0$ space.
Hence the result.
{{qed}}
Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms
Category:Double Pointed Topologies
Category:Finite Complement Topologies
Category:Examples of T0 Spaces
jps2nujhx8sk13qbdfo00qgck4r37la | :$T \times D$ is not a [[Definition:T0 Space|$T_0$ space]] | From [[Double Pointed Space is not T0|Double Pointed Space is not $T_0$]]:
:$T \times D$ is not a [[Definition:T0 Space|$T_0$ space]].
Hence the result.
{{qed}}
[[Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms]]
[[Category:Double Pointed Topologies]]
[[Category:Finite Complement Topol... | Double Pointed Finite Complement Space is not T0 | https://proofwiki.org/wiki/Double_Pointed_Finite_Complement_Space_is_not_T0 | https://proofwiki.org/wiki/Double_Pointed_Finite_Complement_Space_is_not_T0 | [
"Double Pointed Finite Complement Topology fulfils no Separation Axioms",
"Double Pointed Topologies",
"Finite Complement Topologies",
"Examples of T0 Spaces"
] | [
"Definition:T0 Space"
] | [
"Double Pointed Space is not T0",
"Definition:T0 Space",
"Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms",
"Category:Double Pointed Topologies",
"Category:Finite Complement Topologies",
"Category:Examples of T0 Spaces"
] |
proofwiki-23311 | Double Pointed Space is not T1 | 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 $T_1$ space. | {{AimForCont}} $T$ is a $T_1$ space.
From $T_1$ Space is $T_0$ it follows that $T$ is a $T_0$ space.
But from Double Pointed Space is not $T_0$, $T$ is not a $T_0$ space.
Hence the result by Proof by Contradiction.
{{qed}} | 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... | {{AimForCont}} $T$ is a [[Definition:T1 Space|$T_1$ space]].
From [[T1 Space is T0|$T_1$ Space is $T_0$]] it follows that $T$ is a [[Definition:T0 Space|$T_0$ space]].
But from [[Double Pointed Space is not T0|Double Pointed Space is not $T_0$]], $T$ is not a [[Definition:T0 Space|$T_0$ space]].
Hence the result by ... | Double Pointed Space is not T1/Proof 1 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T1 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T1/Proof_1 | [
"Double Pointed Space is not T1",
"Separation Axioms on Double Pointed Topology",
"Double Pointed Topologies",
"Examples of T1 Spaces"
] | [
"Definition:Topological Space",
"Definition:Indiscrete Topology",
"Definition:Arbitrary",
"Definition:Doubleton",
"Definition:Double Pointed Topology",
"Definition:T1 Space"
] | [
"Definition:T1 Space",
"T1 Space is T0",
"Definition:T0 Space",
"Double Pointed Space is not T0",
"Definition:T0 Space",
"Proof by Contradiction"
] |
proofwiki-23312 | Double Pointed Space is not T1 | 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 $T_1$ 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 $T_1$ space.
Then from Product Space is $T_1$ iff Factor Spaces are $T_1$ it follows that $D$ is also a $T_1$ 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 T1/Proof 2 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T1 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T1/Proof_2 | [
"Double Pointed Space is not T1",
"Separation Axioms on Double Pointed Topology",
"Double Pointed Topologies",
"Examples of T1 Spaces"
] | [
"Definition:Topological Space",
"Definition:Indiscrete Topology",
"Definition:Arbitrary",
"Definition:Doubleton",
"Definition:Double Pointed Topology",
"Definition:T1 Space"
] | [
"Definition:Double Pointed Topology",
"Definition:Product Topology",
"Definition:Indiscrete Topology",
"Definition:Doubleton",
"Definition:T1 Space",
"Product Space is T1 iff Factor Spaces are T1",
"Definition:T1 Space",
"Indiscrete Non-Singleton Space is not T1",
"Definition:T1 Space",
"Proof by ... |
proofwiki-23313 | Indiscrete Non-Singleton Space is not T1 | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space which has more than one element.
Then $T$ is not a $T_1$ space. | {{AimForCont}} $T$ is a $T_1$ space.
From $T_1$ Space is $T_0$ Space it follows that $T$ is a $T_0$ space.
But from Indiscrete Non-Singleton Space is not $T_0$, $T$ is not a $T_0$ space.
Hence the result by Proof by Contradiction.
{{qed}}
Category:Indiscrete Topology
Category:Examples of T1 Spaces
gw5tazfr65gvdblkcanm3... | 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:T1 Space|$T_1$ space]]. | {{AimForCont}} $T$ is a [[Definition:T1 Space|$T_1$ space]].
From [[T1 Space is T0 Space|$T_1$ Space is $T_0$ Space]] it follows that $T$ is a [[Definition:T0 Space|$T_0$ space]].
But from [[Indiscrete Non-Singleton Space is not T0|Indiscrete Non-Singleton Space is not $T_0$]], $T$ is not a [[Definition:T0 Space|$T_0... | Indiscrete Non-Singleton Space is not T1 | https://proofwiki.org/wiki/Indiscrete_Non-Singleton_Space_is_not_T1 | https://proofwiki.org/wiki/Indiscrete_Non-Singleton_Space_is_not_T1 | [
"Indiscrete Topology",
"Examples of T1 Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:Element",
"Definition:T1 Space"
] | [
"Definition:T1 Space",
"T1 Space is T0",
"Definition:T0 Space",
"Indiscrete Non-Singleton Space is not T0",
"Definition:T0 Space",
"Proof by Contradiction",
"Category:Indiscrete Topology",
"Category:Examples of T1 Spaces"
] |
proofwiki-23314 | Indiscrete Non-Singleton Space is not T2 | Let $T = \struct {S, \set {\O, S} }$ be an indiscrete topological space which has more than one element.
Then $T$ is not a $T_2$ space. | {{AimForCont}} $T$ is a $T_2$ space.
From $T_2$ Space is $T_1$ Space it follows that $T$ is not a $T_1$ space.
But from Indiscrete Non-Singleton Space is not $T_1$, $T$ is not a $T_1$ space.
Hence the result by Proof by Contradiction.
{{qed}}
Category:Indiscrete Topology
Category:Examples of Hausdorff Spaces
7ixbsfulkh... | 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:T2 Space|$T_2$ space]]. | {{AimForCont}} $T$ is a [[Definition:T2 Space|$T_2$ space]].
From [[T2 Space is T1 Space|$T_2$ Space is $T_1$ Space]] it follows that $T$ is not a [[Definition:T1 Space|$T_1$ space]].
But from [[Indiscrete Non-Singleton Space is not T1|Indiscrete Non-Singleton Space is not $T_1$]], $T$ is not a [[Definition:T1 Space|... | Indiscrete Non-Singleton Space is not T2 | https://proofwiki.org/wiki/Indiscrete_Non-Singleton_Space_is_not_T2 | https://proofwiki.org/wiki/Indiscrete_Non-Singleton_Space_is_not_T2 | [
"Indiscrete Topology",
"Examples of Hausdorff Spaces"
] | [
"Definition:Indiscrete Topology",
"Definition:Element",
"Definition:T2 Space"
] | [
"Definition:T2 Space",
"T2 Space is T1",
"Definition:T1 Space",
"Indiscrete Non-Singleton Space is not T1",
"Definition:T1 Space",
"Proof by Contradiction",
"Category:Indiscrete Topology",
"Category:Examples of Hausdorff Spaces"
] |
proofwiki-23315 | Double Pointed Space is not T2 | 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 $T_2$ (Hausdorff) space. | {{AimForCont}} $T$ is a $T_2$ space.
From $T_2$ Space is $T_1$ Space it follows that $T$ is a $T_1$ space.
But from Double Pointed Space is not $T_1$, $T$ is not a $T_1$ space.
Hence the result by Proof by Contradiction.
{{qed}} | 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... | {{AimForCont}} $T$ is a [[Definition:T2 Space|$T_2$ space]].
From [[T2 Space is T1 Space|$T_2$ Space is $T_1$ Space]] it follows that $T$ is a [[Definition:T1 Space|$T_1$ space]].
But from [[Double Pointed Space is not T1|Double Pointed Space is not $T_1$]], $T$ is not a [[Definition:T1 Space|$T_1$ space]].
Hence th... | Double Pointed Space is not T2/Proof 1 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T2 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T2/Proof_1 | [
"Double Pointed Space is not T2",
"Separation Axioms on Double Pointed Topology",
"Double Pointed Topologies",
"Examples of Hausdorff Spaces"
] | [
"Definition:Topological Space",
"Definition:Indiscrete Topology",
"Definition:Arbitrary",
"Definition:Doubleton",
"Definition:Double Pointed Topology",
"Definition:T2 Space"
] | [
"Definition:T2 Space",
"T2 Space is T1",
"Definition:T1 Space",
"Double Pointed Space is not T1",
"Definition:T1 Space",
"Proof by Contradiction"
] |
proofwiki-23316 | Double Pointed Space is not T2 | 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 $T_2$ (Hausdorff) 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 $T_2$ space.
Then from Product Space is $T_2$ iff Factor Spaces are $T_2$ it follows that $D$ is also a $T_2$ 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 T2/Proof 2 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T2 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T2/Proof_2 | [
"Double Pointed Space is not T2",
"Separation Axioms on Double Pointed Topology",
"Double Pointed Topologies",
"Examples of Hausdorff Spaces"
] | [
"Definition:Topological Space",
"Definition:Indiscrete Topology",
"Definition:Arbitrary",
"Definition:Doubleton",
"Definition:Double Pointed Topology",
"Definition:T2 Space"
] | [
"Definition:Double Pointed Topology",
"Definition:Product Topology",
"Definition:Indiscrete Topology",
"Definition:Doubleton",
"Definition:T2 Space",
"Product Space is T2 iff Factor Spaces are T2",
"Definition:T2 Space",
"Indiscrete Non-Singleton Space is not T2",
"Definition:T2 Space",
"Proof by ... |
proofwiki-23317 | Double Pointed Finite Complement Space is not T1 | :$T \times D$ is not a $T_1$ space | From Double Pointed Space is not $T_1$:
:$T \times D$ is not a $T_1$ space.
Hence the result.
{{qed}}
Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms
Category:Double Pointed Topologies
Category:Finite Complement Topologies
Category:Examples of T1 Spaces
hzs8t7s1kj2yxue6dru3f1hf8sr0ybf | :$T \times D$ is not a [[Definition:T1 Space|$T_1$ space]] | From [[Double Pointed Space is not T1|Double Pointed Space is not $T_1$]]:
:$T \times D$ is not a [[Definition:T1 Space|$T_1$ space]].
Hence the result.
{{qed}}
[[Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms]]
[[Category:Double Pointed Topologies]]
[[Category:Finite Complement Topol... | Double Pointed Finite Complement Space is not T1 | https://proofwiki.org/wiki/Double_Pointed_Finite_Complement_Space_is_not_T1 | https://proofwiki.org/wiki/Double_Pointed_Finite_Complement_Space_is_not_T1 | [
"Double Pointed Finite Complement Topology fulfils no Separation Axioms",
"Double Pointed Topologies",
"Finite Complement Topologies",
"Examples of T1 Spaces"
] | [
"Definition:T1 Space"
] | [
"Double Pointed Space is not T1",
"Definition:T1 Space",
"Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms",
"Category:Double Pointed Topologies",
"Category:Finite Complement Topologies",
"Category:Examples of T1 Spaces"
] |
proofwiki-23318 | Double Pointed Finite Complement Space is not T2 | :$T \times D$ is not a $T_2$ space | From Double Pointed Space is not $T_2$:
:$T \times D$ is not a $T_2$ space.
Hence the result.
{{qed}}
Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms
Category:Double Pointed Topologies
Category:Finite Complement Topologies
Category:Examples of Hausdorff Spaces
rmpzrhkxwzsx0hc9ymkdk65hty9... | :$T \times D$ is not a [[Definition:T2 Space|$T_2$ space]] | From [[Double Pointed Space is not T2|Double Pointed Space is not $T_2$]]:
:$T \times D$ is not a [[Definition:T2 Space|$T_2$ space]].
Hence the result.
{{qed}}
[[Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms]]
[[Category:Double Pointed Topologies]]
[[Category:Finite Complement Topol... | Double Pointed Finite Complement Space is not T2 | https://proofwiki.org/wiki/Double_Pointed_Finite_Complement_Space_is_not_T2 | https://proofwiki.org/wiki/Double_Pointed_Finite_Complement_Space_is_not_T2 | [
"Double Pointed Finite Complement Topology fulfils no Separation Axioms",
"Double Pointed Topologies",
"Finite Complement Topologies",
"Examples of Hausdorff Spaces"
] | [
"Definition:T2 Space"
] | [
"Double Pointed Space is not T2",
"Definition:T2 Space",
"Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms",
"Category:Double Pointed Topologies",
"Category:Finite Complement Topologies",
"Category:Examples of Hausdorff Spaces"
] |
proofwiki-23319 | Double Pointed Finite Complement Space is not T3 | :$T \times D$ is not a $T_3$ space | From Double Pointed Space is $T_3$ iff Factor Space is $T_3$:
:$T \times D$ is a $T_3$ space {{iff}} $T$ is a $T_3$ space
But from Finite Complement Space is not $T_3$:
:$T$ is not a $T_3$ space
Hence the result.
{{qed}}
Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms
Category:Double Poi... | :$T \times D$ is not a [[Definition:T3 Space|$T_3$ space]] | From [[Double Pointed Space is T3 iff Factor Space is T3|Double Pointed Space is $T_3$ iff Factor Space is $T_3$]]:
:$T \times D$ is a [[Definition:T3 Space|$T_3$ space]] {{iff}} $T$ is a [[Definition:T3 Space|$T_3$ space]]
But from [[Finite Complement Space is not T3|Finite Complement Space is not $T_3$]]:
:$T$ is n... | Double Pointed Finite Complement Space is not T3 | https://proofwiki.org/wiki/Double_Pointed_Finite_Complement_Space_is_not_T3 | https://proofwiki.org/wiki/Double_Pointed_Finite_Complement_Space_is_not_T3 | [
"Double Pointed Finite Complement Topology fulfils no Separation Axioms",
"Double Pointed Topologies",
"Finite Complement Topologies",
"Examples of T3 Spaces"
] | [
"Definition:T3 Space"
] | [
"Double Pointed Space is T3 iff Factor Space is T3",
"Definition:T3 Space",
"Definition:T3 Space",
"Finite Complement Space is not T3",
"Definition:T3 Space",
"Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms",
"Category:Double Pointed Topologies",
"Category:Finite Compl... |
proofwiki-23320 | Finite Complement Space is T0 | Let $T = \struct {S, \tau}$ be a finite complement topology on any set $S$ which contains at least two points.
Then $T$ is a $T_0$ space. | We have:
:Finite Complement Space is $T_1$
:$T_1$ Space is $T_0$ Space
Hence the result.
{{qed}}
Category:Finite Complement Topologies
Category:Examples of T0 Spaces
95n25qi4hfva6fuhotfvrxhwp6ce3um | Let $T = \struct {S, \tau}$ be a [[Definition:Finite Complement Topology|finite complement topology]] on any [[Definition:Set|set]] $S$ which contains at least two [[Definition:Point of Set|points]].
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | We have:
:[[Finite Complement Space is T1|Finite Complement Space is $T_1$]]
:[[T1 Space is T0 Space|$T_1$ Space is $T_0$ Space]]
Hence the result.
{{qed}}
[[Category:Finite Complement Topologies]]
[[Category:Examples of T0 Spaces]]
95n25qi4hfva6fuhotfvrxhwp6ce3um | Finite Complement Space is T0 | https://proofwiki.org/wiki/Finite_Complement_Space_is_T0 | https://proofwiki.org/wiki/Finite_Complement_Space_is_T0 | [
"Finite Complement Topologies",
"Examples of T0 Spaces"
] | [
"Definition:Finite Complement Topology",
"Definition:Set",
"Definition:Element",
"Definition:T0 Space"
] | [
"Finite Complement Space is T1",
"T1 Space is T0",
"Category:Finite Complement Topologies",
"Category:Examples of T0 Spaces"
] |
proofwiki-23321 | Finite Complement Space is not T3 | Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$.
Then $T$ is not a $T_3$ space. | Let $S$ be an infinite set.
Let $T = \struct {S, \tau}$ be the finite complement space on $S$.
We have that:
:Finite Complement Space is $T_0$
{{AimForCont}} $T$ is a $T_3$ space.
From Space which is $T_3$ and $T_0$ is also $T_2$, $T$ is a $T_2$ space.
But from Finite Complement Space is not $T_2$, $T$ is not a $T_2$ s... | Let $T = \struct {S, \tau}$ be a [[Definition:Finite Complement Topology|finite complement topology]] on an [[Definition:Infinite Set|infinite set]] $S$.
Then $T$ is not a [[Definition:T3 Space|$T_3$ space]]. | Let $S$ be an [[Definition:Infinite Set|infinite set]].
Let $T = \struct {S, \tau}$ be the [[Definition:Finite Complement Space|finite complement space]] on $S$.
We have that:
:[[Finite Complement Space is T0|Finite Complement Space is $T_0$]]
{{AimForCont}} $T$ is a [[Definition:T3 Space|$T_3$ space]].
From [[Spa... | Finite Complement Space is not T3 | https://proofwiki.org/wiki/Finite_Complement_Space_is_not_T3 | https://proofwiki.org/wiki/Finite_Complement_Space_is_not_T3 | [
"Finite Complement Topologies",
"Examples of T3 Spaces"
] | [
"Definition:Finite Complement Topology",
"Definition:Infinite Set",
"Definition:T3 Space"
] | [
"Definition:Infinite Set",
"Definition:Finite Complement Topology",
"Finite Complement Space is T0",
"Definition:T3 Space",
"Space which is T3 and T0 is also T2",
"Definition:T2 Space",
"Finite Complement Space is not T2",
"Definition:T2 Space",
"Definition:Contradiction",
"Definition:T3 Space"
] |
proofwiki-23322 | Finite Complement Space is not T4 | Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$.
Then $T$ is not a $T_4$ space. | We have that:
:Finite Complement Space is $T_1$
:Normal Space is $T_3$.
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
But from Finite Complement Space is not $T_3$, $T$ is not a normal space.
But $T$ is a $T_1$ space which is not a normal space.
So it follows that $T$ is not ... | Let $T = \struct {S, \tau}$ be a [[Definition:Finite Complement Topology|finite complement topology]] on an [[Definition:Infinite Set|infinite]] set $S$.
Then $T$ is not a [[Definition:T4 Space|$T_4$ space]]. | We have that:
:[[Finite Complement Space is T1|Finite Complement Space is $T_1$]]
:[[Normal Space is T3|Normal Space is $T_3$]].
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
But from [[Finite Complement Space is not T3|Finite Complement Space is not $T_3$]], $T$ is not a ... | Finite Complement Space is not T4 | https://proofwiki.org/wiki/Finite_Complement_Space_is_not_T4 | https://proofwiki.org/wiki/Finite_Complement_Space_is_not_T4 | [
"Finite Complement Topologies",
"Examples of T4 Spaces"
] | [
"Definition:Finite Complement Topology",
"Definition:Infinite Set",
"Definition:T4 Space"
] | [
"Finite Complement Space is T1",
"Normal Space is T3",
"Finite Complement Space is not T3",
"Definition:Normal Space",
"Definition:T1 Space",
"Definition:Normal Space",
"Definition:T4 Space"
] |
proofwiki-23323 | Finite Complement Space is not T5 | Let $T = \struct {S, \tau}$ be a finite complement topology on an infinite set $S$.
Then $T$ is not a $T_5$ space. | {{AimForCont}} $T$ is a $T_5$ space.
From $T_5$ Space is $T_4$:
:$T$ is a $T_4$ space.
But from Finite Complement Space is not $T_4$,
:$T$ is not a $T_4$ space.
Hence, by Proof by Contradiction, $T$ is not a $T_5$ space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Finite Complement Topology|finite complement topology]] on an [[Definition:Infinite Set|infinite]] set $S$.
Then $T$ is not a [[Definition:T5 Space|$T_5$ space]]. | {{AimForCont}} $T$ is a [[Definition:T5 Space|$T_5$ space]].
From [[T5 Space is T4|$T_5$ Space is $T_4$]]:
:$T$ is a [[Definition:T4 Space|$T_4$ space]].
But from [[Finite Complement Space is not T4|Finite Complement Space is not $T_4$]],
:$T$ is not a [[Definition:T4 Space|$T_4$ space]].
Hence, by [[Proof by Contra... | Finite Complement Space is not T5 | https://proofwiki.org/wiki/Finite_Complement_Space_is_not_T5 | https://proofwiki.org/wiki/Finite_Complement_Space_is_not_T5 | [
"Finite Complement Topologies",
"Examples of T5 Spaces"
] | [
"Definition:Finite Complement Topology",
"Definition:Infinite Set",
"Definition:T5 Space"
] | [
"Definition:T5 Space",
"T5 Space is T4",
"Definition:T4 Space",
"Finite Complement Space is not T4",
"Definition:T4 Space",
"Proof by Contradiction",
"Definition:T5 Space"
] |
proofwiki-23324 | Double Pointed Space is not T2.5 | Let $T_S = \struct {S, \tau_S}$ be a topological space.
Let $D = \struct {A, \set {\O, A} }$ be the indiscrete space on an arbitrary doubleton $A = \set {a, b}$.
Let $T = \struct {T_S \times D, \tau}$ be the double pointed topological space on $T_S$.
Then $T$ is not a $T_{2 \frac 1 2}$ space. | {{AimForCont}} $T$ is a $T_{2 \frac 1 2}$ space.
From $T_{2 \frac 1 2}$ Space is $T_2$ it follows that $T$ is a $T_2$ space.
But from Double Pointed Space is not $T_2$, $T$ is not a $T_2$ space.
Hence the result by Proof by Contradiction.
{{qed}}
Category:Separation Axioms on Double Pointed Topology
Category:Double Poi... | 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... | {{AimForCont}} $T$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]].
From [[T2.5 Space is T2|$T_{2 \frac 1 2}$ Space is $T_2$]] it follows that $T$ is a [[Definition:T2 Space|$T_2$ space]].
But from [[Double Pointed Space is not T2|Double Pointed Space is not $T_2$]], $T$ is not a [[Definition:T2 Space|$T_2$ sp... | Double Pointed Space is not T2.5 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T2.5 | https://proofwiki.org/wiki/Double_Pointed_Space_is_not_T2.5 | [
"Separation Axioms on Double Pointed Topology",
"Double Pointed Topologies",
"Examples of T2.5 Spaces"
] | [
"Definition:Topological Space",
"Definition:Indiscrete Topology",
"Definition:Arbitrary",
"Definition:Doubleton",
"Definition:Double Pointed Topology",
"Definition:T2.5 Space"
] | [
"Definition:T2.5 Space",
"T2.5 Space is T2",
"Definition:T2 Space",
"Double Pointed Space is not T2",
"Definition:T2 Space",
"Proof by Contradiction",
"Category:Separation Axioms on Double Pointed Topology",
"Category:Double Pointed Topologies",
"Category:Examples of T2.5 Spaces"
] |
proofwiki-23325 | Double Pointed Finite Complement Space is not T4 | :$T \times D$ is not a $T_4$ space | From Double Pointed Space is $T_4$ iff Factor Space is $T_4$ :
:$T \times D$ is a $T_4$ space {{iff}} $T$ is a $T_4$ space
But from Finite Complement Space is not $T_4$:
:$T$ is not a $T_4$ space
Hence the result.
{{qed}}
Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms
Category:Double Po... | :$T \times D$ is not a [[Definition:T4 Space|$T_4$ space]] | From [[Double Pointed Space is T4 iff Factor Space is T4 |Double Pointed Space is $T_4$ iff Factor Space is $T_4$ ]]:
:$T \times D$ is a [[Definition:T4 Space|$T_4$ space]] {{iff}} $T$ is a [[Definition:T4 Space|$T_4$ space]]
But from [[Finite Complement Space is not T4|Finite Complement Space is not $T_4$]]:
:$T$ is... | Double Pointed Finite Complement Space is not T4 | https://proofwiki.org/wiki/Double_Pointed_Finite_Complement_Space_is_not_T4 | https://proofwiki.org/wiki/Double_Pointed_Finite_Complement_Space_is_not_T4 | [
"Double Pointed Finite Complement Topology fulfils no Separation Axioms",
"Double Pointed Topologies",
"Finite Complement Topologies",
"Examples of T4 Spaces"
] | [
"Definition:T4 Space"
] | [
"Double Pointed Space is T4 iff Factor Space is T4 ",
"Definition:T4 Space",
"Definition:T4 Space",
"Finite Complement Space is not T4",
"Definition:T4 Space",
"Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms",
"Category:Double Pointed Topologies",
"Category:Finite Comp... |
proofwiki-23326 | Double Pointed Finite Complement Space is not T5 | :$T \times D$ is not a $T_5$ space | From Double Pointed Space is $T_5$ iff Factor Space is $T_5$:
:$T \times D$ is a $T_5$ space {{iff}} $T$ is a $T_5$ space
But from Finite Complement Space is not $T_5$:
:$T$ is not a $T_5$ space
Hence the result.
{{qed}}
Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms
Category:Double Poi... | :$T \times D$ is not a [[Definition:T5 Space|$T_5$ space]] | From [[Double Pointed Space is T5 iff Factor Space is T5|Double Pointed Space is $T_5$ iff Factor Space is $T_5$]]:
:$T \times D$ is a [[Definition:T5 Space|$T_5$ space]] {{iff}} $T$ is a [[Definition:T5 Space|$T_5$ space]]
But from [[Finite Complement Space is not T5|Finite Complement Space is not $T_5$]]:
:$T$ is n... | Double Pointed Finite Complement Space is not T5 | https://proofwiki.org/wiki/Double_Pointed_Finite_Complement_Space_is_not_T5 | https://proofwiki.org/wiki/Double_Pointed_Finite_Complement_Space_is_not_T5 | [
"Double Pointed Finite Complement Topology fulfils no Separation Axioms",
"Double Pointed Topologies",
"Finite Complement Topologies",
"Examples of T5 Spaces"
] | [
"Definition:T5 Space"
] | [
"Double Pointed Space is T5 iff Factor Space is T5",
"Definition:T5 Space",
"Definition:T5 Space",
"Finite Complement Space is not T5",
"Definition:T5 Space",
"Category:Double Pointed Finite Complement Topology fulfils no Separation Axioms",
"Category:Double Pointed Topologies",
"Category:Finite Compl... |
proofwiki-23327 | Irrational Slope Space is not T3 | Let $T$ be an irrational slope space.
Then $T$ is not a $T_3$ space. | {{AimForCont}} $T$ is a $T_3$ space.
We have:
:Irrational Slope Space is $T_0$
{{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
So, as $T$ is both a $T_3$ space and a $T_0$ space, it follows that $T$ is a regular space.
But from Irrational Slope Space is not Regular, $T$ is not a regular sp... | Let $T$ be an [[Definition:Irrational Slope Space|irrational slope space]].
Then $T$ is not a [[Definition:T3 Space|$T_3$ space]]. | {{AimForCont}} $T$ is a [[Definition:T3 Space|$T_3$ space]].
We have:
:[[Irrational Slope Space is T0|Irrational Slope Space is $T_0$]]
{{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
So, as $T$ is both a [[Definition:T3 Space|$T_3$ space]] and a [[Definition:T0 Space|$T_0$ space]], it... | Irrational Slope Space is not T3 | https://proofwiki.org/wiki/Irrational_Slope_Space_is_not_T3 | https://proofwiki.org/wiki/Irrational_Slope_Space_is_not_T3 | [
"Irrational Slope Topology",
"Examples of T3 Spaces"
] | [
"Definition:Irrational Slope Topology",
"Definition:T3 Space"
] | [
"Definition:T3 Space",
"Irrational Slope Space is T0",
"Definition:T3 Space",
"Definition:T0 Space",
"Definition:Regular Space",
"Irrational Slope Space is not Regular",
"Definition:Regular Space",
"Definition:Contradiction",
"Definition:Assumption",
"Definition:T3 Space",
"Proof by Contradictio... |
proofwiki-23328 | Irrational Slope Space is not Regular | Let $T$ be an irrational slope space.
Then $T$ is not a regular space. | {{AimForCont}} $T$ is a regular space.
From Regular Space is $T_{2 \frac 1 2}$, $T$ is a $T_{2 \frac 1 2}$ space.
But from Irrational Slope Space is not $T_{2 \frac 1 2}$, $T$ is not a $T_{2 \frac 1 2}$ space.
From that contradiction it follows that our assumption that $T$ is a regular space must be false.
Hence the re... | Let $T$ be an [[Definition:Irrational Slope Space|irrational slope space]].
Then $T$ is not a [[Definition:Regular Space|regular space]]. | {{AimForCont}} $T$ is a [[Definition:Regular Space|regular space]].
From [[Regular Space is T2.5|Regular Space is $T_{2 \frac 1 2}$]], $T$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]].
But from [[Irrational Slope Space is not T2.5|Irrational Slope Space is not $T_{2 \frac 1 2}$]], $T$ is not a [[Definition:... | Irrational Slope Space is not Regular | https://proofwiki.org/wiki/Irrational_Slope_Space_is_not_Regular | https://proofwiki.org/wiki/Irrational_Slope_Space_is_not_Regular | [
"Irrational Slope Topology",
"Examples of Regular Spaces"
] | [
"Definition:Irrational Slope Topology",
"Definition:Regular Space"
] | [
"Definition:Regular Space",
"Regular Space is T2.5",
"Definition:T2.5 Space",
"Irrational Slope Space is not T2.5",
"Definition:T2.5 Space",
"Definition:Contradiction",
"Definition:Assumption",
"Definition:Regular Space",
"Proof by Contradiction",
"Category:Irrational Slope Topology",
"Category:... |
proofwiki-23329 | Irrational Slope Space is not Normal | Let $T$ be an irrational slope space.
Then $T$ is not a normal space. | {{AimForCont}} $T$ is a normal space.
Hence from Normal Space is Regular we have that $T$ is a regular space.
But from Irrational Slope Space is not Regular, $T$ is not a regular space.
From that contradiction it follows that our assumption that $T$ is a normal space must be false.
Hence the result by Proof by Contradi... | Let $T$ be an [[Definition:Irrational Slope Space|irrational slope space]].
Then $T$ is not a [[Definition:Normal Space|normal space]]. | {{AimForCont}} $T$ is a [[Definition:Normal Space|normal space]].
Hence from [[Normal Space is Regular]] we have that $T$ is a [[Definition:Regular Space|regular space]].
But from [[Irrational Slope Space is not Regular]], $T$ is not a [[Definition:Regular Space|regular space]].
From that [[Definition:Contradiction|... | Irrational Slope Space is not Normal | https://proofwiki.org/wiki/Irrational_Slope_Space_is_not_Normal | https://proofwiki.org/wiki/Irrational_Slope_Space_is_not_Normal | [
"Irrational Slope Topology",
"Examples of Normal Spaces"
] | [
"Definition:Irrational Slope Topology",
"Definition:Normal Space"
] | [
"Definition:Normal Space",
"Normal Space is Regular",
"Definition:Regular Space",
"Irrational Slope Space is not Regular",
"Definition:Regular Space",
"Definition:Contradiction",
"Definition:Assumption",
"Definition:Normal Space",
"Proof by Contradiction",
"Category:Irrational Slope Topology",
"... |
proofwiki-23330 | Irrational Slope Space is not T4 | Let $T$ be an irrational slope space.
Then $T$ is not a $T_4$ space. | {{AimForCont}} $T$ is a $T_4$ space.
We have:
:Irrational Slope Space is $T_1$.
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
So, as $T$ is both a $T_4$ space and a $T_1$ space, it follows that $T$ is a normal space.
But from Irrational Slope Space is not Normal, $T$ is not a ... | Let $T$ be an [[Definition:Irrational Slope Space|irrational slope space]].
Then $T$ is not a [[Definition:T4 Space|$T_4$ space]]. | {{AimForCont}} $T$ is a [[Definition:T4 Space|$T_4$ space]].
We have:
:[[Irrational Slope Space is T1|Irrational Slope Space is $T_1$]].
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
So, as $T$ is both a [[Definition:T4 Space|$T_4$ space]] and a [[Definition:T1 Space|$T_1$... | Irrational Slope Space is not T4 | https://proofwiki.org/wiki/Irrational_Slope_Space_is_not_T4 | https://proofwiki.org/wiki/Irrational_Slope_Space_is_not_T4 | [
"Irrational Slope Topology",
"Examples of T4 Spaces"
] | [
"Definition:Irrational Slope Topology",
"Definition:T4 Space"
] | [
"Definition:T4 Space",
"Irrational Slope Space is T1",
"Definition:T4 Space",
"Definition:T1 Space",
"Definition:Normal Space",
"Irrational Slope Space is not Normal",
"Definition:Normal Space",
"Definition:Contradiction",
"Definition:Assumption",
"Definition:T4 Space",
"Proof by Contradiction"
... |
proofwiki-23331 | Irrational Slope Space is T1 | Let $T$ be an irrational slope space.
Then $T$ is a $T_1$ space. | We have:
:Irrational Slope Space is $T_2$
:$T_2$ Space is $T_1$
Hence the result.
{{qed}}
Category:Irrational Slope Topology
Category:Examples of T1 Spaces
7ommf4sa0hfiwseevey0ettu7e400ck | Let $T$ be an [[Definition:Irrational Slope Space|irrational slope space]].
Then $T$ is a [[Definition:T1 Space|$T_1$ space]]. | We have:
:[[Irrational Slope Space is T2|Irrational Slope Space is $T_2$]]
:[[T2 Space is T1|$T_2$ Space is $T_1$]]
Hence the result.
{{qed}}
[[Category:Irrational Slope Topology]]
[[Category:Examples of T1 Spaces]]
7ommf4sa0hfiwseevey0ettu7e400ck | Irrational Slope Space is T1 | https://proofwiki.org/wiki/Irrational_Slope_Space_is_T1 | https://proofwiki.org/wiki/Irrational_Slope_Space_is_T1 | [
"Irrational Slope Topology",
"Examples of T1 Spaces"
] | [
"Definition:Irrational Slope Topology",
"Definition:T1 Space"
] | [
"Irrational Slope Space is T2",
"T2 Space is T1",
"Category:Irrational Slope Topology",
"Category:Examples of T1 Spaces"
] |
proofwiki-23332 | Irrational Slope Space is T0 | Let $T$ be an irrational slope space.
Then $T$ is a $T_0$ space. | We have:
:Irrational Slope Space is $T_1$
:$T_1$ Space is $T_0$
Hence the result.
{{qed}}
Category:Irrational Slope Topology
Category:Examples of T0 Spaces
f862y3qgbngkj5ny27x61wtythsiesn | Let $T$ be an [[Definition:Irrational Slope Space|irrational slope space]].
Then $T$ is a [[Definition:T0 Space|$T_0$ space]]. | We have:
:[[Irrational Slope Space is T1|Irrational Slope Space is $T_1$]]
:[[T1 Space is T0|$T_1$ Space is $T_0$]]
Hence the result.
{{qed}}
[[Category:Irrational Slope Topology]]
[[Category:Examples of T0 Spaces]]
f862y3qgbngkj5ny27x61wtythsiesn | Irrational Slope Space is T0 | https://proofwiki.org/wiki/Irrational_Slope_Space_is_T0 | https://proofwiki.org/wiki/Irrational_Slope_Space_is_T0 | [
"Irrational Slope Topology",
"Examples of T0 Spaces"
] | [
"Definition:Irrational Slope Topology",
"Definition:T0 Space"
] | [
"Irrational Slope Space is T1",
"T1 Space is T0",
"Category:Irrational Slope Topology",
"Category:Examples of T0 Spaces"
] |
proofwiki-23333 | Irrational Slope Space is not T5 | Let $T$ be an irrational slope space.
Then $T$ is not a $T_5$ space. | {{AimForCont}} $T$ is a $T_5$ space.
We have:
:$T_5$ Space is $T_4$.
But from Irrational Slope Space is not $T_4$, $T$ is not a $T_4$ space.
From that contradiction it follows that our assumption that $T$ is a $T_5$ space must be false.
Hence the result by Proof by Contradiction.
{{qed}} | Let $T$ be an [[Definition:Irrational Slope Space|irrational slope space]].
Then $T$ is not a [[Definition:T5 Space|$T_5$ space]]. | {{AimForCont}} $T$ is a [[Definition:T5 Space|$T_5$ space]].
We have:
:[[T5 Space is T4|$T_5$ Space is $T_4$]].
But from [[Irrational Slope Space is not T4|Irrational Slope Space is not $T_4$]], $T$ is not a [[Definition:T4 Space|$T_4$ space]].
From that [[Definition:Contradiction|contradiction]] it follows that our... | Irrational Slope Space is not T5 | https://proofwiki.org/wiki/Irrational_Slope_Space_is_not_T5 | https://proofwiki.org/wiki/Irrational_Slope_Space_is_not_T5 | [
"Irrational Slope Topology",
"Examples of T5 Spaces"
] | [
"Definition:Irrational Slope Topology",
"Definition:T5 Space"
] | [
"Definition:T5 Space",
"T5 Space is T4",
"Irrational Slope Space is not T4",
"Definition:T4 Space",
"Definition:Contradiction",
"Definition:Assumption",
"Definition:T5 Space",
"Proof by Contradiction"
] |
proofwiki-23334 | Tychonoff Plank is not T5 | Let $T$ be a Tychonoff plank.
Then $T$ is not a $T_5$ space. | Let $T$ be the Tychonoff plank.
{{AimForCont}} $T$ is a $T_5$ space.
{{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
From Tychonoff Plank is $T_1$, $T$ is a $T_1$ space.
Hence $T$ is a completely normal space.
But from Tychonoff Plank is not Comp... | Let $T$ be a [[Definition:Tychonoff Plank|Tychonoff plank]].
Then $T$ is not a [[Definition:T5 Space|$T_5$ space]]. | Let $T$ be the [[Definition:Tychonoff Plank|Tychonoff plank]].
{{AimForCont}} $T$ is a [[Definition:T5 Space|$T_5$ space]].
{{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
From [[Tychonoff Plank is T1|Tychonoff Plank is $T_1$]], $T$ is a [[De... | Tychonoff Plank is not T5 | https://proofwiki.org/wiki/Tychonoff_Plank_is_not_T5 | https://proofwiki.org/wiki/Tychonoff_Plank_is_not_T5 | [
"Tychonoff Plank",
"Examples of T5 Spaces"
] | [
"Definition:Tychonoff Plank",
"Definition:T5 Space"
] | [
"Definition:Tychonoff Plank",
"Definition:T5 Space",
"Tychonoff Plank is T1",
"Definition:T1 Space",
"Definition:Completely Normal Space",
"Tychonoff Plank is not Completely Normal",
"Definition:Completely Normal Space",
"Proof by Contradiction",
"Category:Tychonoff Plank",
"Category:Examples of T... |
proofwiki-23335 | Tychonoff Plank is T1 | Let $T$ be a Tychonoff plank.
Then $T$ is a $T_1$ space. | From Tychonoff Plank is $T_2$, $T$ is a $T_2$ space.
The result follows from $T_2$ Space is $T_1$.
{{qed}}
Category:Tychonoff Plank
Category:Examples of T1 Spaces
6725ygg1zixgookipjbt4tn5r6h1mcq | Let $T$ be a [[Definition:Tychonoff Plank|Tychonoff plank]].
Then $T$ is a [[Definition:T1 Space|$T_1$ space]]. | From [[Tychonoff Plank is T2|Tychonoff Plank is $T_2$]], $T$ is a [[Definition:T2 Space|$T_2$ space]].
The result follows from [[T2 Space is T1|$T_2$ Space is $T_1$]].
{{qed}}
[[Category:Tychonoff Plank]]
[[Category:Examples of T1 Spaces]]
6725ygg1zixgookipjbt4tn5r6h1mcq | Tychonoff Plank is T1 | https://proofwiki.org/wiki/Tychonoff_Plank_is_T1 | https://proofwiki.org/wiki/Tychonoff_Plank_is_T1 | [
"Tychonoff Plank",
"Examples of T1 Spaces"
] | [
"Definition:Tychonoff Plank",
"Definition:T1 Space"
] | [
"Tychonoff Plank is T2",
"Definition:T2 Space",
"T2 Space is T1",
"Category:Tychonoff Plank",
"Category:Examples of T1 Spaces"
] |
proofwiki-23336 | Uncountable Cartesian Product of Closed Unit Interval is Compact Space | Let $T = \Bbb I^\Bbb I$ be the uncountable Cartesian product of the closed unit interval under the usual (Euclidean) topology.
Then $T$ is a compact space. | We have that the Closed Real Interval is Compact Space.
{{ProofWanted|There should be something about a product space of compact spaces is compact somewhere}} | Let $T = \Bbb I^\Bbb I$ be the [[Definition:Uncountable Cartesian Product of Closed Unit Interval|uncountable Cartesian product of the closed unit interval]] under the [[Definition:Euclidean Topology on Real Number Line|usual (Euclidean) topology]].
Then $T$ is a [[Definition:Compact Topological Space|compact space]]. | We have that the [[Closed Real Interval is Compact Space]].
{{ProofWanted|There should be something about a product space of compact spaces is compact somewhere}} | Uncountable Cartesian Product of Closed Unit Interval is Compact Space | https://proofwiki.org/wiki/Uncountable_Cartesian_Product_of_Closed_Unit_Interval_is_Compact_Space | https://proofwiki.org/wiki/Uncountable_Cartesian_Product_of_Closed_Unit_Interval_is_Compact_Space | [
"Uncountable Cartesian Product of Closed Unit Interval",
"Examples of Compact Topological Spaces"
] | [
"Definition:Uncountable Cartesian Product of Closed Unit Interval",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Compact Topological Space"
] | [
"Closed Real Interval is Compact Space"
] |
proofwiki-23337 | Sequentially Compact Space is not necessarily Compact | A sequentially compact space is not necessarily also a compact space. | Let $\hointr 0 \Omega$ denote the uncountable open ordinal space under the order topology.
From Uncountable Open Ordinal Space is Sequentially Compact Space, $\hointr 0 \Omega$ is a sequentially compact space.
From Open Ordinal Space is not Compact Space, $\hointr 0 \Omega$ is not a compact space.
Hence the result.
{{q... | A [[Definition:Sequentially Compact Space|sequentially compact space]] is not necessarily also a [[Definition:Compact Topological Space|compact space]]. | Let $\hointr 0 \Omega$ denote the [[Definition:Uncountable Open Ordinal Space|uncountable open ordinal space]] under the [[Definition:Order Topology|order topology]].
From [[Uncountable Open Ordinal Space is Sequentially Compact Space]], $\hointr 0 \Omega$ is a [[Definition:Sequentially Compact Space|sequentially com... | Sequentially Compact Space is not necessarily Compact | https://proofwiki.org/wiki/Sequentially_Compact_Space_is_not_necessarily_Compact | https://proofwiki.org/wiki/Sequentially_Compact_Space_is_not_necessarily_Compact | [
"Sequentially Compact Spaces",
"Compact Topological Spaces",
"Sequence of Implications of Global Compactness Properties"
] | [
"Definition:Sequentially Compact Space",
"Definition:Compact Topological Space"
] | [
"Definition:Ordinal Space/Open/Uncountable",
"Definition:Order Topology",
"Uncountable Open Ordinal Space is Sequentially Compact Space",
"Definition:Sequentially Compact Space",
"Open Ordinal Space is not Compact Space",
"Definition:Compact Topological Space"
] |
proofwiki-23338 | Countably Compact Space is not necessarily Compact | A countably compact topological space is not necessarily also a compact space. | Let $T := \hointr 0 \Omega \times \Bbb I^\Bbb I$ be the Cartesian product of $\hointr 0 \Omega$ under the interval topology, with the uncountable Cartesian product of the closed unit interval under the usual (Euclidean) topology.
From Cartesian Product of Open Ordinal Space with Uncountable Cartesian Product of Closed ... | A [[Definition:Countably Compact Space|countably compact topological space]] is not necessarily also a [[Definition:Compact Topological Space|compact space]]. | Let $T := \hointr 0 \Omega \times \Bbb I^\Bbb I$ be the [[Definition:Cartesian Product|Cartesian product]] of $\hointr 0 \Omega$ under the [[Definition:Interval Topology|interval topology]], with the [[Definition:Uncountable Set|uncountable]] [[Definition:Cartesian Product|Cartesian product]] of the [[Definition:Closed... | Countably Compact Space is not necessarily Compact/Proof 1 | https://proofwiki.org/wiki/Countably_Compact_Space_is_not_necessarily_Compact | https://proofwiki.org/wiki/Countably_Compact_Space_is_not_necessarily_Compact/Proof_1 | [
"Countably Compact Space is not necessarily Compact",
"Countably Compact Spaces",
"Compact Topological Spaces"
] | [
"Definition:Countably Compact Space",
"Definition:Compact Topological Space"
] | [
"Definition:Cartesian Product",
"Definition:Order Topology",
"Definition:Uncountable/Set",
"Definition:Cartesian Product",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Cartesian Product of Open Ordinal Space with Uncountable Cartesia... |
proofwiki-23339 | Strongly Locally Compact Space is not necessarily Weakly Sigma-Locally Compact | A strongly locally compact space is not necessarily also a weakly $\sigma$-locally compact topological space. | Let $T$ be an uncountable discrete space.
From Discrete Space is Strongly Locally Compact, $T$ is a strongly locally compact space.
From Uncountable Discrete Space is not Weakly $\sigma$-Locally Compact, $T$ is not a weakly $\sigma$-locally compact space.
Hence the result.
{{qed}} | A [[Definition:Strongly Locally Compact Space|strongly locally compact space]] is not necessarily also a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact topological space]]. | Let $T$ be an [[Definition:Uncountable Discrete Space|uncountable discrete space]].
From [[Discrete Space is Strongly Locally Compact]], $T$ is a [[Definition:Strongly Locally Compact Space|strongly locally compact space]].
From [[Uncountable Discrete Space is not Weakly Sigma-Locally Compact|Uncountable Discrete Sp... | Strongly Locally Compact Space is not necessarily Weakly Sigma-Locally Compact | https://proofwiki.org/wiki/Strongly_Locally_Compact_Space_is_not_necessarily_Weakly_Sigma-Locally_Compact | https://proofwiki.org/wiki/Strongly_Locally_Compact_Space_is_not_necessarily_Weakly_Sigma-Locally_Compact | [
"Strongly Locally Compact Spaces",
"Weakly Sigma-Locally Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Strongly Locally Compact Space",
"Definition:Weakly Sigma-Locally Compact Space"
] | [
"Definition:Discrete Topology/Uncountable",
"Discrete Space is Strongly Locally Compact",
"Definition:Strongly Locally Compact Space",
"Uncountable Discrete Space is not Weakly Sigma-Locally Compact",
"Definition:Weakly Sigma-Locally Compact Space"
] |
proofwiki-23340 | Uncountable Discrete Space is not Weakly Sigma-Locally Compact | Let $T = \struct {S, \tau}$ be an uncountable discrete topological space.
Then $T$ is not a weakly $\sigma$-locally compact space. | {{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
{{:Definition:Weakly Sigma-Locally Compact Space}}
We have that an Uncountable Discrete Space is not $\sigma$-Compact.
But a weakly $\sigma$-locally compact space is a $\sigma$-compact space by definition.
So an uncountable discrete spa... | Let $T = \struct {S, \tau}$ be an [[Definition:Uncountable Discrete Space|uncountable discrete topological space]].
Then $T$ is not a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space]]. | {{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
{{:Definition:Weakly Sigma-Locally Compact Space}}
We have that an [[Uncountable Discrete Space is not Sigma-Compact|Uncountable Discrete Space is not $\sigma$-Compact]].
But a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\... | Uncountable Discrete Space is not Weakly Sigma-Locally Compact | https://proofwiki.org/wiki/Uncountable_Discrete_Space_is_not_Weakly_Sigma-Locally_Compact | https://proofwiki.org/wiki/Uncountable_Discrete_Space_is_not_Weakly_Sigma-Locally_Compact | [
"Discrete Topologies",
"Examples of Weakly Sigma-Locally Compact Spaces",
"Uncountable Sets"
] | [
"Definition:Discrete Topology/Uncountable",
"Definition:Weakly Sigma-Locally Compact Space"
] | [
"Uncountable Discrete Space is not Sigma-Compact",
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:Sigma-Compact Space",
"Definition:Discrete Topology/Uncountable",
"Definition:Weakly Sigma-Locally Compact Space",
"Category:Discrete Topologies",
"Category:Examples of Weakly Sigma-Locally Co... |
proofwiki-23341 | Weakly Locally Compact Space is not necessarily Weakly Sigma-Locally Compact | A weakly locally compact topological space is not necessarily also a weakly $\sigma$-locally compact space. | Let $T$ be an uncountable particular point space.
From Particular Point Space is Weakly Locally Compact, $T$ is a weakly locally compact space.
From Uncountable Particular Point Space is not Weakly Sigma-Locally Compact, $T$ is not a weakly $\sigma$-locally compact space.
Hence the result.
{{qed}} | A [[Definition:Weakly Locally Compact Space|weakly locally compact topological space]] is not necessarily also a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space]]. | Let $T$ be an [[Definition:Uncountable Particular Point Space|uncountable particular point space]].
From [[Particular Point Space is Weakly Locally Compact]], $T$ is a [[Definition:Weakly Locally Compact Space|weakly locally compact space]].
From [[Uncountable Particular Point Space is not Weakly Sigma-Locally Compa... | Weakly Locally Compact Space is not necessarily Weakly Sigma-Locally Compact/Proof 1 | https://proofwiki.org/wiki/Weakly_Locally_Compact_Space_is_not_necessarily_Weakly_Sigma-Locally_Compact | https://proofwiki.org/wiki/Weakly_Locally_Compact_Space_is_not_necessarily_Weakly_Sigma-Locally_Compact/Proof_1 | [
"Weakly Locally Compact Space is not necessarily Weakly Sigma-Locally Compact",
"Weakly Locally Compact Spaces",
"Weakly Sigma-Locally Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:Weakly Sigma-Locally Compact Space"
] | [
"Definition:Particular Point Topology/Uncountable",
"Particular Point Space is Weakly Locally Compact",
"Definition:Weakly Locally Compact Space",
"Uncountable Particular Point Space is not Weakly Sigma-Locally Compact",
"Definition:Weakly Sigma-Locally Compact Space"
] |
proofwiki-23342 | Weakly Locally Compact Space is not necessarily Weakly Sigma-Locally Compact | A weakly locally compact topological space is not necessarily also a weakly $\sigma$-locally compact space. | Let $T$ be an uncountable discrete space.
From Discrete Space is Weakly Locally Compact, $T$ is a weakly locally compact space.
From Uncountable Discrete Space is not Weakly Sigma-Locally Compact, $T$ is not a weakly $\sigma$-locally compact space.
Hence the result.
{{qed}} | A [[Definition:Weakly Locally Compact Space|weakly locally compact topological space]] is not necessarily also a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space]]. | Let $T$ be an [[Definition:Uncountable Discrete Space|uncountable discrete space]].
From [[Discrete Space is Weakly Locally Compact]], $T$ is a [[Definition:Weakly Locally Compact Space|weakly locally compact space]].
From [[Uncountable Discrete Space is not Weakly Sigma-Locally Compact]], $T$ is not a [[Definition:... | Weakly Locally Compact Space is not necessarily Weakly Sigma-Locally Compact/Proof 2 | https://proofwiki.org/wiki/Weakly_Locally_Compact_Space_is_not_necessarily_Weakly_Sigma-Locally_Compact | https://proofwiki.org/wiki/Weakly_Locally_Compact_Space_is_not_necessarily_Weakly_Sigma-Locally_Compact/Proof_2 | [
"Weakly Locally Compact Space is not necessarily Weakly Sigma-Locally Compact",
"Weakly Locally Compact Spaces",
"Weakly Sigma-Locally Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:Weakly Sigma-Locally Compact Space"
] | [
"Definition:Discrete Topology/Uncountable",
"Discrete Space is Weakly Locally Compact",
"Definition:Weakly Locally Compact Space",
"Uncountable Discrete Space is not Weakly Sigma-Locally Compact",
"Definition:Weakly Sigma-Locally Compact Space"
] |
proofwiki-23343 | Uncountable Particular Point Space is not Sigma-Compact | Let $T = \struct {S, \tau}$ be an uncountable particular point space.
Then $T$ is not a $\sigma$-compact space. | We have that an Uncountable Particular Point Space is not Lindelöf.
But a $\sigma$-compact space is a Lindelöf space.
So an uncountable particular point space cannot be $\sigma$-compact.
{{qed}}
Category:Particular Point Topologies
Category:Examples of Sigma-Compact Spaces
Category:Uncountable Sets
kjubvlzejbo68zzhd0f8... | Let $T = \struct {S, \tau}$ be an [[Definition:Uncountable Particular Point Space|uncountable particular point space]].
Then $T$ is not a [[Definition:Sigma-Compact Space|$\sigma$-compact space]]. | We have that an [[Uncountable Particular Point Space is not Lindelöf]].
But a [[Sigma-Compact Space is Lindelöf Space|$\sigma$-compact space is a Lindelöf space]].
So an [[Definition:Uncountable Particular Point Space|uncountable particular point space]] cannot be [[Definition:Sigma-Compact Space|$\sigma$-compact]].
... | Uncountable Particular Point Space is not Sigma-Compact | https://proofwiki.org/wiki/Uncountable_Particular_Point_Space_is_not_Sigma-Compact | https://proofwiki.org/wiki/Uncountable_Particular_Point_Space_is_not_Sigma-Compact | [
"Particular Point Topologies",
"Examples of Sigma-Compact Spaces",
"Uncountable Sets"
] | [
"Definition:Particular Point Topology/Uncountable",
"Definition:Sigma-Compact Space"
] | [
"Uncountable Particular Point Space is not Lindelöf",
"Sigma-Compact Space is Lindelöf",
"Definition:Particular Point Topology/Uncountable",
"Definition:Sigma-Compact Space",
"Category:Particular Point Topologies",
"Category:Examples of Sigma-Compact Spaces",
"Category:Uncountable Sets"
] |
proofwiki-23344 | Uncountable Particular Point Space is not Weakly Sigma-Locally Compact | Let $T = \struct {S, \tau}$ be an uncountable particular point space.
Then $T$ is not a weakly $\sigma$-locally compact space. | {{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
{{:Definition:Weakly Sigma-Locally Compact Space}}
We have that an Uncountable Particular Point Space is not $\sigma$-Compact.
But a weakly $\sigma$-locally compact space is a $\sigma$-compact space by definition.
So an uncountable part... | Let $T = \struct {S, \tau}$ be an [[Definition:Uncountable Particular Point Space|uncountable particular point space]].
Then $T$ is not a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space]]. | {{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
{{:Definition:Weakly Sigma-Locally Compact Space}}
We have that an [[Uncountable Particular Point Space is not Sigma-Compact|Uncountable Particular Point Space is not $\sigma$-Compact]].
But a [[Definition:Weakly Sigma-Locally Compact... | Uncountable Particular Point Space is not Weakly Sigma-Locally Compact | https://proofwiki.org/wiki/Uncountable_Particular_Point_Space_is_not_Weakly_Sigma-Locally_Compact | https://proofwiki.org/wiki/Uncountable_Particular_Point_Space_is_not_Weakly_Sigma-Locally_Compact | [
"Particular Point Topologies",
"Examples of Weakly Sigma-Locally Compact Spaces",
"Uncountable Sets"
] | [
"Definition:Particular Point Topology/Uncountable",
"Definition:Weakly Sigma-Locally Compact Space"
] | [
"Uncountable Particular Point Space is not Sigma-Compact",
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:Sigma-Compact Space",
"Definition:Particular Point Topology/Uncountable",
"Definition:Weakly Sigma-Locally Compact Space",
"Category:Particular Point Topologies",
"Category:Examples of... |
proofwiki-23345 | Discrete Space is Weakly Locally Compact | Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is weakly locally compact. | From Discrete Space is Strongly Locally Compact, $T$ is a strongly locally compact space.
Hence the result from Strongly Locally Compact Space is Weakly Locally Compact.
{{qed}}
Category:Discrete Topologies
Category:Examples of Weakly Locally Compact Spaces
4bx3gkeqe0p5xad1cmdg4ygw6fuafhz | Let $T = \struct {S, \tau}$ be a [[Definition:Discrete Space|discrete topological space]].
Then $T$ is [[Definition:Weakly Locally Compact Space|weakly locally compact]]. | From [[Discrete Space is Strongly Locally Compact]], $T$ is a [[Definition:Strongly Locally Compact Space|strongly locally compact space]].
Hence the result from [[Strongly Locally Compact Space is Weakly Locally Compact]].
{{qed}}
[[Category:Discrete Topologies]]
[[Category:Examples of Weakly Locally Compact Spaces]... | Discrete Space is Weakly Locally Compact | https://proofwiki.org/wiki/Discrete_Space_is_Weakly_Locally_Compact | https://proofwiki.org/wiki/Discrete_Space_is_Weakly_Locally_Compact | [
"Discrete Topologies",
"Examples of Weakly Locally Compact Spaces"
] | [
"Definition:Discrete Topology",
"Definition:Weakly Locally Compact Space"
] | [
"Discrete Space is Strongly Locally Compact",
"Definition:Strongly Locally Compact Space",
"Strongly Locally Compact Space is Weakly Locally Compact",
"Category:Discrete Topologies",
"Category:Examples of Weakly Locally Compact Spaces"
] |
proofwiki-23346 | Weakly Locally Compact Space is not necessarily Sigma-Compact | A weakly locally compact topological space is not necessarily also a $\sigma$-compact space. | Let $T$ be an uncountable discrete space.
From Discrete Space is Weakly Locally Compact, $T$ is a weakly locally compact space.
From Uncountable Discrete Space is not $\sigma$-Compact, $T$ is not a $\sigma$-compact space.
Hence the result.
{{qed}}
Category:Weakly Locally Compact Spaces
Category:Sigma-Compact Spaces
Cat... | A [[Definition:Weakly Locally Compact Space|weakly locally compact topological space]] is not necessarily also a [[Definition:Sigma-Compact Space|$\sigma$-compact space]]. | Let $T$ be an [[Definition:Uncountable Discrete Space|uncountable discrete space]].
From [[Discrete Space is Weakly Locally Compact]], $T$ is a [[Definition:Weakly Locally Compact Space|weakly locally compact space]].
From [[Uncountable Discrete Space is not Sigma-Compact|Uncountable Discrete Space is not $\sigma$-C... | Weakly Locally Compact Space is not necessarily Sigma-Compact | https://proofwiki.org/wiki/Weakly_Locally_Compact_Space_is_not_necessarily_Sigma-Compact | https://proofwiki.org/wiki/Weakly_Locally_Compact_Space_is_not_necessarily_Sigma-Compact | [
"Weakly Locally Compact Spaces",
"Sigma-Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:Sigma-Compact Space"
] | [
"Definition:Discrete Topology/Uncountable",
"Discrete Space is Weakly Locally Compact",
"Definition:Weakly Locally Compact Space",
"Uncountable Discrete Space is not Sigma-Compact",
"Definition:Sigma-Compact Space",
"Category:Weakly Locally Compact Spaces",
"Category:Sigma-Compact Spaces",
"Category:S... |
proofwiki-23347 | Weakly Locally Compact Space is not necessarily Lindelöf | A weakly locally compact topological space is not necessarily also a Lindelöf space. | Let $T$ be an uncountable discrete space.
From Discrete Space is Weakly Locally Compact, $T$ is a weakly locally compact space.
From Uncountable Discrete Space is not Lindelöf, $T$ is not a Lindelöf space.
Hence the result.
{{qed}}
Category:Weakly Locally Compact Spaces
Category:Lindelöf Spaces
Category:Sequence of Imp... | A [[Definition:Weakly Locally Compact Space|weakly locally compact topological space]] is not necessarily also a [[Definition:Lindelöf Space|Lindelöf space]]. | Let $T$ be an [[Definition:Uncountable Discrete Space|uncountable discrete space]].
From [[Discrete Space is Weakly Locally Compact]], $T$ is a [[Definition:Weakly Locally Compact Space|weakly locally compact space]].
From [[Uncountable Discrete Space is not Lindelöf]], $T$ is not a [[Definition:Lindelöf Space|Linde... | Weakly Locally Compact Space is not necessarily Lindelöf | https://proofwiki.org/wiki/Weakly_Locally_Compact_Space_is_not_necessarily_Lindelöf | https://proofwiki.org/wiki/Weakly_Locally_Compact_Space_is_not_necessarily_Lindelöf | [
"Weakly Locally Compact Spaces",
"Lindelöf Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:Lindelöf Space"
] | [
"Definition:Discrete Topology/Uncountable",
"Discrete Space is Weakly Locally Compact",
"Definition:Weakly Locally Compact Space",
"Uncountable Discrete Space is not Lindelöf",
"Definition:Lindelöf Space",
"Category:Weakly Locally Compact Spaces",
"Category:Lindelöf Spaces",
"Category:Sequence of Impl... |
proofwiki-23348 | Weakly Sigma-Locally Compact Space is not necessarily Compact | A weakly $\sigma$-locally compact topological space is not necessarily also a compact space. | Let $T$ be the nested interval topological space.
From Nested Interval Space is Weakly $\sigma$-Locally Compact, $T$ is a weakly $\sigma$-locally compact space.
From Nested Interval Space is not Compact, $T$ is not a compact space.
Hence the result.
{{qed}}
Category:Compact Topological Spaces
Category:Weakly Sigma-Loca... | A [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact topological space]] is not necessarily also a [[Definition:Compact Topological Space|compact space]]. | Let $T$ be the [[Definition:Nested Interval Space|nested interval topological space]].
From [[Nested Interval Space is Weakly Sigma-Locally Compact|Nested Interval Space is Weakly $\sigma$-Locally Compact]], $T$ is a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space]].
From [[Nest... | Weakly Sigma-Locally Compact Space is not necessarily Compact | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_Space_is_not_necessarily_Compact | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_Space_is_not_necessarily_Compact | [
"Compact Topological Spaces",
"Weakly Sigma-Locally Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:Compact Topological Space"
] | [
"Definition:Nested Interval Topology",
"Nested Interval Space is Weakly Sigma-Locally Compact",
"Definition:Weakly Sigma-Locally Compact Space",
"Nested Interval Space is not Compact",
"Definition:Compact Topological Space",
"Category:Compact Topological Spaces",
"Category:Weakly Sigma-Locally Compact S... |
proofwiki-23349 | Nested Interval Space is not Compact | Let $T = \struct {S, \tau}$ be a nested interval space.
Then $T$ is ''not'' a compact space. | We have Nested Interval Space is not Strongly Locally Compact.
From Compact Space is Strongly Locally Compact, it follows that $T$ cannot be a compact space.
{{qed}}
Category:Nested Interval Topology
Category:Examples of Compact Topological Spaces
ocyfb8fu6nlxk0wj06drasthuorsb0x | Let $T = \struct {S, \tau}$ be a [[Definition:Nested Interval Space|nested interval space]].
Then $T$ is ''not'' a [[Definition:Compact Topological Space|compact space]]. | We have [[Nested Interval Space is not Strongly Locally Compact]].
From [[Compact Space is Strongly Locally Compact]], it follows that $T$ cannot be a [[Definition:Compact Topological Space|compact space]].
{{qed}}
[[Category:Nested Interval Topology]]
[[Category:Examples of Compact Topological Spaces]]
ocyfb8fu6nlxk... | Nested Interval Space is not Compact | https://proofwiki.org/wiki/Nested_Interval_Space_is_not_Compact | https://proofwiki.org/wiki/Nested_Interval_Space_is_not_Compact | [
"Nested Interval Topology",
"Examples of Compact Topological Spaces"
] | [
"Definition:Nested Interval Topology",
"Definition:Compact Topological Space"
] | [
"Nested Interval Space is not Strongly Locally Compact",
"Compact Space is Strongly Locally Compact",
"Definition:Compact Topological Space",
"Category:Nested Interval Topology",
"Category:Examples of Compact Topological Spaces"
] |
proofwiki-23350 | Strongly Locally Compact Space is not necessarily Lindelöf | A strongly locally compact topological space is not necessarily also a Lindelöf space. | Let $T$ be an uncountable discrete space.
From Discrete Space is Strongly Locally Compact, $T$ is a strongly locally compact space.
From Uncountable Discrete Space is not Lindelöf, $T$ is not a Lindelöf space.
Hence the result.
{{qed}}
Category:Strongly Locally Compact Spaces
Category:Lindelöf Spaces
Category:Sequence ... | A [[Definition:Strongly Locally Compact Space|strongly locally compact topological space]] is not necessarily also a [[Definition:Lindelöf Space|Lindelöf space]]. | Let $T$ be an [[Definition:Uncountable Discrete Topology|uncountable discrete space]].
From [[Discrete Space is Strongly Locally Compact]], $T$ is a [[Definition:Strongly Locally Compact Space|strongly locally compact space]].
From [[Uncountable Discrete Space is not Lindelöf]], $T$ is not a [[Definition:Lindelöf Sp... | Strongly Locally Compact Space is not necessarily Lindelöf | https://proofwiki.org/wiki/Strongly_Locally_Compact_Space_is_not_necessarily_Lindelöf | https://proofwiki.org/wiki/Strongly_Locally_Compact_Space_is_not_necessarily_Lindelöf | [
"Strongly Locally Compact Spaces",
"Lindelöf Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Strongly Locally Compact Space",
"Definition:Lindelöf Space"
] | [
"Definition:Discrete Topology/Uncountable",
"Discrete Space is Strongly Locally Compact",
"Definition:Strongly Locally Compact Space",
"Uncountable Discrete Space is not Lindelöf",
"Definition:Lindelöf Space",
"Category:Strongly Locally Compact Spaces",
"Category:Lindelöf Spaces",
"Category:Sequence o... |
proofwiki-23351 | Strongly Locally Compact Space is not necessarily Compact | A strongly locally compact topological space is not necessarily also a compact space. | Let $T$ be an infinite discrete space.
From Discrete Space is Strongly Locally Compact, $T$ is a strongly locally compact space.
From Discrete Space is Compact iff Finite, $T$ is not a compact space.
Hence the result.
{{qed}}
Category:Strongly Locally Compact Spaces
Category:Compact Topological Spaces
Category:Sequence... | A [[Definition:Strongly Locally Compact Space|strongly locally compact topological space]] is not necessarily also a [[Definition:Compact Topological Space|compact space]]. | Let $T$ be an [[Definition:Infinite Discrete Space|infinite discrete space]].
From [[Discrete Space is Strongly Locally Compact]], $T$ is a [[Definition:Strongly Locally Compact Space|strongly locally compact space]].
From [[Discrete Space is Compact iff Finite]], $T$ is not a [[Definition:Compact Topological Space|... | Strongly Locally Compact Space is not necessarily Compact | https://proofwiki.org/wiki/Strongly_Locally_Compact_Space_is_not_necessarily_Compact | https://proofwiki.org/wiki/Strongly_Locally_Compact_Space_is_not_necessarily_Compact | [
"Strongly Locally Compact Spaces",
"Compact Topological Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Strongly Locally Compact Space",
"Definition:Compact Topological Space"
] | [
"Definition:Discrete Topology/Infinite",
"Discrete Space is Strongly Locally Compact",
"Definition:Strongly Locally Compact Space",
"Discrete Space is Compact iff Finite",
"Definition:Compact Topological Space",
"Category:Strongly Locally Compact Spaces",
"Category:Compact Topological Spaces",
"Catego... |
proofwiki-23352 | Sigma-Compact Space is not necessarily Weakly Sigma-Locally Compact | A $\sigma$-compact space is not necessarily also a weakly $\sigma$-locally compact space. | Let $T$ be the evenly spaced integer space.
From Evenly Spaced Integer Space is $\sigma$-Compact, $T$ is a $\sigma$-compact space.
From Evenly Spaced Integer Space is not Weakly $\sigma$-Locally Compact, $T$ is not a weakly $\sigma$-locally compact space.
Hence the result.
{{qed}}
Category:Sigma-Compact Spaces
Category... | A [[Definition:Sigma-Compact Space|$\sigma$-compact space]] is not necessarily also a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space]]. | Let $T$ be the [[Definition:Evenly Spaced Integer Space|evenly spaced integer space]].
From [[Evenly Spaced Integer Space is Sigma-Compact|Evenly Spaced Integer Space is $\sigma$-Compact]], $T$ is a [[Definition:Sigma-Compact Space|$\sigma$-compact space]].
From [[Evenly Spaced Integer Space is not Weakly Sigma-Loca... | Sigma-Compact Space is not necessarily Weakly Sigma-Locally Compact | https://proofwiki.org/wiki/Sigma-Compact_Space_is_not_necessarily_Weakly_Sigma-Locally_Compact | https://proofwiki.org/wiki/Sigma-Compact_Space_is_not_necessarily_Weakly_Sigma-Locally_Compact | [
"Sigma-Compact Spaces",
"Weakly Sigma-Locally Compact Spaces",
"Sequence of Implications of Local Compactness Properties"
] | [
"Definition:Sigma-Compact Space",
"Definition:Weakly Sigma-Locally Compact Space"
] | [
"Definition:Evenly Spaced Integer Topology",
"Evenly Spaced Integer Space is Sigma-Compact",
"Definition:Sigma-Compact Space",
"Evenly Spaced Integer Space is not Weakly Sigma-Locally Compact",
"Definition:Weakly Sigma-Locally Compact Space",
"Category:Sigma-Compact Spaces",
"Category:Weakly Sigma-Local... |
proofwiki-23353 | Evenly Spaced Integer Space is Sigma-Compact | The evenly spaced integer space is a $\sigma$-compact space. | We have Evenly Spaced Integer Space is Countably Infinite.
The result follows from Countable Space is $\sigma$-Compact.
{{qed}} | The [[Definition:Evenly Spaced Integer Space|evenly spaced integer space]] is a [[Definition:Sigma-Compact Space|$\sigma$-compact space]]. | We have [[Evenly Spaced Integer Space is Countably Infinite]].
The result follows from [[Countable Space is Sigma-Compact|Countable Space is $\sigma$-Compact]].
{{qed}} | Evenly Spaced Integer Space is Sigma-Compact | https://proofwiki.org/wiki/Evenly_Spaced_Integer_Space_is_Sigma-Compact | https://proofwiki.org/wiki/Evenly_Spaced_Integer_Space_is_Sigma-Compact | [
"Evenly Spaced Integer Topology",
"Examples of Sigma-Compact Spaces"
] | [
"Definition:Evenly Spaced Integer Topology",
"Definition:Sigma-Compact Space"
] | [
"Evenly Spaced Integer Space is Countably Infinite",
"Countable Space is Sigma-Compact"
] |
proofwiki-23354 | Evenly Spaced Integer Space is Countably Infinite | The evenly spaced integer space is a countably infinite topological space. | By its definition, $S$ is the set of integers $\Z$.
The result follows from Integers are Countably Infinite.
{{qed}} | The [[Definition:Evenly Spaced Integer Space|evenly spaced integer space]] is a [[Definition:Countably Infinite Topological Space|countably infinite topological space]]. | By its definition, $S$ is the [[Definition:Integer|set of integers]] $\Z$.
The result follows from [[Integers are Countably Infinite]].
{{qed}} | Evenly Spaced Integer Space is Countably Infinite | https://proofwiki.org/wiki/Evenly_Spaced_Integer_Space_is_Countably_Infinite | https://proofwiki.org/wiki/Evenly_Spaced_Integer_Space_is_Countably_Infinite | [
"Evenly Spaced Integer Topology",
"Examples of Countably Infinite Topological Spaces"
] | [
"Definition:Evenly Spaced Integer Topology",
"Definition:Countably Infinite Topological Space"
] | [
"Definition:Integer",
"Integers are Countably Infinite"
] |
proofwiki-23355 | Evenly Spaced Integer Space is not Weakly Locally Compact | The evenly spaced integer space is not a weakly locally compact space. | {{Recall|Weakly Locally Compact Space|weakly locally compact space}}
{{:Definition:Weakly Locally Compact Space}}
{{ProofWanted}} | The [[Definition:Evenly Spaced Integer Space|evenly spaced integer space]] is not a [[Definition:Weakly Locally Compact Space|weakly locally compact space]]. | {{Recall|Weakly Locally Compact Space|weakly locally compact space}}
{{:Definition:Weakly Locally Compact Space}}
{{ProofWanted}} | Evenly Spaced Integer Space is not Weakly Locally Compact | https://proofwiki.org/wiki/Evenly_Spaced_Integer_Space_is_not_Weakly_Locally_Compact | https://proofwiki.org/wiki/Evenly_Spaced_Integer_Space_is_not_Weakly_Locally_Compact | [
"Evenly Spaced Integer Topology",
"Examples of Weakly Locally Compact Spaces"
] | [
"Definition:Evenly Spaced Integer Topology",
"Definition:Weakly Locally Compact Space"
] | [] |
proofwiki-23356 | Evenly Spaced Integer Space is not Weakly Sigma-Locally Compact | The evenly spaced integer space is not a weakly $\sigma$-locally compact space. | {{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
{{:Definition:Weakly Sigma-Locally Compact Space}}
We have Evenly Spaced Integer Space is not Weakly Locally Compact.
Hence as $T$ is not weakly locally compact, it cannot be weakly $\sigma$-locally compact.
Hence the result.
{{qed}}
Ca... | The [[Definition:Evenly Spaced Integer Space|evenly spaced integer space]] is not a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space]]. | {{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
{{:Definition:Weakly Sigma-Locally Compact Space}}
We have [[Evenly Spaced Integer Space is not Weakly Locally Compact]].
Hence as $T$ is not [[Definition:Weakly Locally Compact Space|weakly locally compact]], it cannot be [[Definitio... | Evenly Spaced Integer Space is not Weakly Sigma-Locally Compact | https://proofwiki.org/wiki/Evenly_Spaced_Integer_Space_is_not_Weakly_Sigma-Locally_Compact | https://proofwiki.org/wiki/Evenly_Spaced_Integer_Space_is_not_Weakly_Sigma-Locally_Compact | [
"Evenly Spaced Integer Topology",
"Examples of Weakly Sigma-Locally Compact Spaces"
] | [
"Definition:Evenly Spaced Integer Topology",
"Definition:Weakly Sigma-Locally Compact Space"
] | [
"Evenly Spaced Integer Space is not Weakly Locally Compact",
"Definition:Weakly Locally Compact Space",
"Definition:Weakly Sigma-Locally Compact Space",
"Category:Evenly Spaced Integer Topology",
"Category:Examples of Weakly Sigma-Locally Compact Spaces"
] |
proofwiki-23357 | Dieudonné Plank is not Paracompact | Let $T = \struct {S, \tau}$ be the Dieudonné plank.
Then $T$ is not paracompact. | We have that Dieudonné Plank is not Countably Paracompact.
From Paracompact Space is Countably Paracompact it follows that $T$ cannot be paracompact.
{{qed}}
Category:Dieudonné Plank
Category:Examples of Paracompact Spaces
f8hpm7oqtm61bqyzt3o8nrvk2iduq1n | Let $T = \struct {S, \tau}$ be the [[Definition:Dieudonné Plank|Dieudonné plank]].
Then $T$ is not [[Definition:Paracompact Space|paracompact]]. | We have that [[Dieudonné Plank is not Countably Paracompact]].
From [[Paracompact Space is Countably Paracompact]] it follows that $T$ cannot be [[Definition:Paracompact Space|paracompact]].
{{qed}}
[[Category:Dieudonné Plank]]
[[Category:Examples of Paracompact Spaces]]
f8hpm7oqtm61bqyzt3o8nrvk2iduq1n | Dieudonné Plank is not Paracompact | https://proofwiki.org/wiki/Dieudonné_Plank_is_not_Paracompact | https://proofwiki.org/wiki/Dieudonné_Plank_is_not_Paracompact | [
"Dieudonné Plank",
"Dieudonné Plank",
"Examples of Paracompact Spaces"
] | [
"Definition:Dieudonné Plank",
"Definition:Paracompact Space"
] | [
"Dieudonné Plank is not Countably Paracompact",
"Paracompact Space is Countably Paracompact",
"Definition:Paracompact Space",
"Category:Dieudonné Plank",
"Category:Examples of Paracompact Spaces"
] |
proofwiki-23358 | Metacompact Space is not necessarily Countably Paracompact | A metacompact topological space is not necessarily also a paracompact space. | Let $T$ be the Dieudonné plank.
From Dieudonné Plank is Metacompact, $T$ is a metacompact space.
From Dieudonné Plank is not Countably Paracompact, $T$ is not a countably paracompact space.
Hence the result.
{{qed}}
Category:Metacompact Spaces
Category:Countably Paracompact Spaces
Category:Sequence of Implications of P... | A [[Definition:Metacompact Space|metacompact topological space]] is not necessarily also a [[Definition:Paracompact Space|paracompact space]]. | Let $T$ be the [[Definition:Dieudonné Plank|Dieudonné plank]].
From [[Dieudonné Plank is Metacompact]], $T$ is a [[Definition:Metacompact Space|metacompact space]].
From [[Dieudonné Plank is not Countably Paracompact]], $T$ is not a [[Definition:Countably Paracompact Space|countably paracompact space]].
Hence the r... | Metacompact Space is not necessarily Countably Paracompact | https://proofwiki.org/wiki/Metacompact_Space_is_not_necessarily_Countably_Paracompact | https://proofwiki.org/wiki/Metacompact_Space_is_not_necessarily_Countably_Paracompact | [
"Metacompact Spaces",
"Countably Paracompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Metacompact Space",
"Definition:Paracompact Space"
] | [
"Definition:Dieudonné Plank",
"Dieudonné Plank is Metacompact",
"Definition:Metacompact Space",
"Dieudonné Plank is not Countably Paracompact",
"Definition:Countably Paracompact Space",
"Category:Metacompact Spaces",
"Category:Countably Paracompact Spaces",
"Category:Sequence of Implications of Paraco... |
proofwiki-23359 | Countably Paracompact Space is not necessarily Paracompact | A countably paracompact topological space is not necessarily also a paracompact space. | Let $T = \struct {S, \tau}$ be Michael's closed subspace.
From Michael's Closed Subspace is Countably Paracompact, $T$ is a countably paracompact space.
From Michael's Closed Subspace is not Paracompact, $T$ is not a paracompact space.
Hence the result.
{{qed}} | A [[Definition:Countably Paracompact Space|countably paracompact topological space]] is not necessarily also a [[Definition:Paracompact Space|paracompact space]]. | Let $T = \struct {S, \tau}$ be [[Definition:Michael's Closed Subspace|Michael's closed subspace]].
From [[Michael's Closed Subspace is Countably Paracompact]], $T$ is a [[Definition:Countably Paracompact Space|countably paracompact space]].
From [[Michael's Closed Subspace is not Paracompact]], $T$ is not a [[Defini... | Countably Paracompact Space is not necessarily Paracompact | https://proofwiki.org/wiki/Countably_Paracompact_Space_is_not_necessarily_Paracompact | https://proofwiki.org/wiki/Countably_Paracompact_Space_is_not_necessarily_Paracompact | [
"Countably Paracompact Spaces",
"Paracompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Countably Paracompact Space",
"Definition:Paracompact Space"
] | [
"Definition:Michael's Closed Subspace",
"Michael's Closed Subspace is Countably Paracompact",
"Definition:Countably Paracompact Space",
"Michael's Closed Subspace is not Paracompact",
"Definition:Paracompact Space"
] |
proofwiki-23360 | Countably Metacompact Space is not necessarily Metacompact | A countably metacompact topological space is not necessarily also a metacompact space. | Let $T$ be a half-disc space.
From Half-Disc Space is Countably Metacompact, $T$ is a countably metacompact space.
From Half-Disc Space is not Metacompact, $T$ is not a metacompact space.
Hence the result.
{{qed}}
Category:Countably Metacompact Spaces
Category:Metacompact Spaces
Category:Sequence of Implications of Par... | A [[Definition:Countably Metacompact Space|countably metacompact topological space]] is not necessarily also a [[Definition:Metacompact Space|metacompact space]]. | Let $T$ be a [[Definition:Half-Disc Topology|half-disc space]].
From [[Half-Disc Space is Countably Metacompact]], $T$ is a [[Definition:Countably Metacompact Space|countably metacompact space]].
From [[Half-Disc Space is not Metacompact]], $T$ is not a [[Definition:Metacompact Space|metacompact space]].
Hence the... | Countably Metacompact Space is not necessarily Metacompact | https://proofwiki.org/wiki/Countably_Metacompact_Space_is_not_necessarily_Metacompact | https://proofwiki.org/wiki/Countably_Metacompact_Space_is_not_necessarily_Metacompact | [
"Countably Metacompact Spaces",
"Metacompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Countably Metacompact Space",
"Definition:Metacompact Space"
] | [
"Definition:Half-Disc Topology",
"Half-Disc Space is Countably Metacompact",
"Definition:Countably Metacompact Space",
"Half-Disc Space is not Metacompact",
"Definition:Metacompact Space",
"Category:Countably Metacompact Spaces",
"Category:Metacompact Spaces",
"Category:Sequence of Implications of Par... |
proofwiki-23361 | Countably Compact Space is not necessarily Metacompact | A countably compact topological space is not necessarily also a metacompact space. | Let $\hointr 0 \Omega$ denote the uncountable open ordinal space under the order topology.
From Uncountable Open Ordinal Space is Countably Compact Space, $\hointr 0 \Omega$ is a countably compact space.
From Uncountable Open Ordinal Space is not Metacompact Space, $\hointr 0 \Omega$ is not a metacompact space.
Hence t... | A [[Definition:Countably Compact Space|countably compact topological space]] is not necessarily also a [[Definition:Metacompact Space|metacompact space]]. | Let $\hointr 0 \Omega$ denote the [[Definition:Uncountable Open Ordinal Space|uncountable open ordinal space]] under the [[Definition:Order Topology|order topology]].
From [[Uncountable Open Ordinal Space is Countably Compact Space]], $\hointr 0 \Omega$ is a [[Definition:Countably Compact Space|countably compact spac... | Countably Compact Space is not necessarily Metacompact | https://proofwiki.org/wiki/Countably_Compact_Space_is_not_necessarily_Metacompact | https://proofwiki.org/wiki/Countably_Compact_Space_is_not_necessarily_Metacompact | [
"Countably Compact Spaces",
"Metacompact Spaces",
"Sequence of Implications of Paracompactness Properties"
] | [
"Definition:Countably Compact Space",
"Definition:Metacompact Space"
] | [
"Definition:Ordinal Space/Open/Uncountable",
"Definition:Order Topology",
"Uncountable Open Ordinal Space is Countably Compact Space",
"Definition:Countably Compact Space",
"Uncountable Open Ordinal Space is not Metacompact Space",
"Definition:Metacompact Space",
"Category:Countably Compact Spaces",
"... |
proofwiki-23362 | Refinement of Cover is not necessarily Subcover | Let $S$ be a set.
Let $\CC$ be a cover of $S$.
Let $\RR$ be a refinement of $\CC$.
Then $\RR$ is not necessarily a subcover of $\CC$. | {{Recall|Refinement of Cover|refinement}}
{{:Definition:Refinement of Cover}}
{{Recall|Subcover|subcover}}
{{:Definition:Subcover}}
;Proof by Counterexample
Let $S = \set {1, 2}$.
Let $\CC = \set {\set {1, 2} }$.
Then $\CC$ is a cover of $S$.
Let $\RR = \set {\set 1, \set 2}$.
Then $\RR$ is a refinement of $\CC$.
But n... | Let $S$ be a [[Definition:Set|set]].
Let $\CC$ be a [[Definition:Cover of Set|cover]] of $S$.
Let $\RR$ be a [[Definition:Refinement of Cover|refinement]] of $\CC$.
Then $\RR$ is not necessarily a [[Definition:Subcover|subcover]] of $\CC$. | {{Recall|Refinement of Cover|refinement}}
{{:Definition:Refinement of Cover}}
{{Recall|Subcover|subcover}}
{{:Definition:Subcover}}
;[[Proof by Counterexample]]
Let $S = \set {1, 2}$.
Let $\CC = \set {\set {1, 2} }$.
Then $\CC$ is a [[Definition:Cover of Set|cover]] of $S$.
Let $\RR = \set {\set 1, \set 2}$.
The... | Refinement of Cover is not necessarily Subcover | https://proofwiki.org/wiki/Refinement_of_Cover_is_not_necessarily_Subcover | https://proofwiki.org/wiki/Refinement_of_Cover_is_not_necessarily_Subcover | [
"Refinements of Covers",
"Subcovers",
"Covers"
] | [
"Definition:Set",
"Definition:Cover of Set",
"Definition:Refinement of Cover",
"Definition:Subcover"
] | [
"Proof by Counterexample",
"Definition:Cover of Set",
"Definition:Refinement of Cover",
"Definition:Element"
] |
proofwiki-23363 | Global Compactness Properties which are Weakly Hereditary | The following global compactness properties are weakly hereditary:
:Compactness
:Countable Compactness
{{WIP}} | We have:
* Closed Subspace of Compact Space is Compact
* Countable Compactness is Weakly Hereditary
{{ProofWanted}} | The following [[Definition:Global Compactness Property|global compactness properties]] are [[Definition:Weakly Hereditary Property|weakly hereditary]]:
:[[Definition:Compact Topological Space|Compactness]]
:[[Definition:Countably Compact Space|Countable Compactness]]
{{WIP}} | We have:
* [[Closed Subspace of Compact Space is Compact]]
* [[Countable Compactness is Weakly Hereditary]]
{{ProofWanted}} | Global Compactness Properties which are Weakly Hereditary | https://proofwiki.org/wiki/Global_Compactness_Properties_which_are_Weakly_Hereditary | https://proofwiki.org/wiki/Global_Compactness_Properties_which_are_Weakly_Hereditary | [
"Global Compactness Properties which are Weakly Hereditary",
"Global Compactness Properties",
"Weakly Hereditary Properties"
] | [
"Definition:Global Compactness Property",
"Definition:Weakly Hereditary Property",
"Definition:Compact Topological Space",
"Definition:Countably Compact Space"
] | [
"Closed Subspace of Compact Space is Compact",
"Countable Compactness is Weakly Hereditary"
] |
proofwiki-23364 | Separability is Preserved only in Open Subspace | The property of separability is preserved only in subspaces which are open. | Let $T = \struct {S, \tau}$ be a separable space.
Let $H \subseteq T$ be a subspace of $T$ such that $H \in \tau$.
We are to show that $H$ is a separable space.
{{Recall|Separable Space|separable space}}
{{:Definition:Separable Space}}
{{ProofWanted}}
Now let $K \subseteq T$ be a subspace of $T$ such that $K \notin \ta... | The property of [[Definition:Separable Space|separability]] is preserved only in [[Definition:Topological Subspace|subspaces]] which are [[Definition:Open Set (Topology)|open]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Separable Space|separable space]].
Let $H \subseteq T$ be a [[Definition:Topological Subspace|subspace]] of $T$ such that $H \in \tau$.
We are to show that $H$ is a [[Definition:Separable Space|separable space]].
{{Recall|Separable Space|separable space}}
{{:Definition:... | Separability is Preserved only in Open Subspace | https://proofwiki.org/wiki/Separability_is_Preserved_only_in_Open_Subspace | https://proofwiki.org/wiki/Separability_is_Preserved_only_in_Open_Subspace | [
"Separable Spaces",
"Topological Subspaces"
] | [
"Definition:Separable Space",
"Definition:Topological Subspace",
"Definition:Open Set/Topology"
] | [
"Definition:Separable Space",
"Definition:Topological Subspace",
"Definition:Separable Space",
"Definition:Topological Subspace",
"Definition:Separable Space"
] |
proofwiki-23365 | Second-Countable Regular Space is Completely Normal | Let $T = \struct {S, \tau}$ be a second-countable space which is also regular.
Then $T$ is a completely normal space. | {{Recall|Regular Space|regular space|index = 2}}
{{:Definition:Regular Space/Definition 2}}
{{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
So, let $T$ be a second-countable regular space.
From Second-Countable $T_3$ Space is $T_5$, $T$ is a $T_5... | Let $T = \struct {S, \tau}$ be a [[Definition:Second-Countable Space|second-countable space]] which is also [[Definition:Regular Space|regular]].
Then $T$ is a [[Definition:Completely Normal Space|completely normal space]]. | {{Recall|Regular Space|regular space|index = 2}}
{{:Definition:Regular Space/Definition 2}}
{{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
So, let $T$ be a [[Definition:Second-Countable Space|second-countable]] [[Definition:Regular Space|regul... | Second-Countable Regular Space is Completely Normal | https://proofwiki.org/wiki/Second-Countable_Regular_Space_is_Completely_Normal | https://proofwiki.org/wiki/Second-Countable_Regular_Space_is_Completely_Normal | [
"Second-Countable Spaces",
"Regular Spaces",
"Completely Normal Spaces"
] | [
"Definition:Second-Countable Space",
"Definition:Regular Space",
"Definition:Completely Normal Space"
] | [
"Definition:Second-Countable Space",
"Definition:Regular Space",
"Second-Countable T3 Space is T5",
"Definition:T5 Space",
"Definition:T5 Space",
"Definition:T1 Space"
] |
proofwiki-23366 | T2 Space is Fully T4 iff Paracompact | Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) topological space.
Then:
:$T$ is a fully $T_4$ space
{{iff}}:
:$T$ is a paracompact space. | We are given that $T = \struct {S, \tau}$ is a $T_2$ (Hausdorff) space. | Let $T = \struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) topological space]].
Then:
:$T$ is a [[Definition:Fully T4 Space|fully $T_4$ space]]
{{iff}}:
:$T$ is a [[Definition:Paracompact Space|paracompact space]]. | We are given that $T = \struct {S, \tau}$ is a [[Definition:T2 Space|$T_2$ (Hausdorff) space]]. | T2 Space is Fully T4 iff Paracompact | https://proofwiki.org/wiki/T2_Space_is_Fully_T4_iff_Paracompact | https://proofwiki.org/wiki/T2_Space_is_Fully_T4_iff_Paracompact | [
"Fully T4 Spaces",
"Paracompact Spaces",
"Hausdorff Spaces",
"Sequence of Implications of Compactness Properties in Hausdorff Space"
] | [
"Definition:T2 Space",
"Definition:Fully T4 Space",
"Definition:Paracompact Space"
] | [
"Definition:T2 Space"
] |
proofwiki-23367 | Paracompact T2 Space is Normal | Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space which is paracompact.
Then $T$ is a normal space. | {{Recall|Normal Space|normal space|index = 2}}
{{:Definition:Normal Space/Definition 2}}
Let $T = \struct {S, \tau}$ be a paracompact $T_2$ (Hausdorff) space.
From $T_2$ Space is Fully $T_4$ iff Paracompact, $T$ is a fully $T_4$ space.
From Fully $T_4$ Space is $T_4$ Space, $T$ is a $T_4$ space.
Hence the result.
{{qed... | Let $T = \struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] which is [[Definition:Paracompact Space|paracompact]].
Then $T$ is a [[Definition:Normal Space|normal space]]. | {{Recall|Normal Space|normal space|index = 2}}
{{:Definition:Normal Space/Definition 2}}
Let $T = \struct {S, \tau}$ be a [[Definition:Paracompact Space|paracompact]] [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
From [[T2 Space is Fully T4 iff Paracompact|$T_2$ Space is Fully $T_4$ iff Paracompact]], $T$ is a [[D... | Paracompact T2 Space is Normal | https://proofwiki.org/wiki/Paracompact_T2_Space_is_Normal | https://proofwiki.org/wiki/Paracompact_T2_Space_is_Normal | [
"Paracompact Spaces",
"Normal Spaces",
"Hausdorff Spaces",
"Sequence of Implications of Compactness Properties in Hausdorff Space"
] | [
"Definition:T2 Space",
"Definition:Paracompact Space",
"Definition:Normal Space"
] | [
"Definition:Paracompact Space",
"Definition:T2 Space",
"T2 Space is Fully T4 iff Paracompact",
"Definition:Fully T4 Space",
"Fully T4 Space is T4",
"Definition:T4 Space"
] |
proofwiki-23368 | Weakly Sigma-Locally Compact T2 Space is Normal | Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space which is weakly $\sigma$-locally compact.
Then $T$ is a normal space. | {{Recall|Normal Space|normal space|index = 2}}
{{:Definition:Normal Space/Definition 2}}
Let $T = \struct {S, \tau}$ be a weakly $\sigma$-locally compact $T_2$ (Hausdorff) space.
From Weakly $\sigma$-Locally Compact $T_2$ Space is $T_4$, $T$ is a $T_4$ space.
Hence the result.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] which is [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact]].
Then $T$ is a [[Definition:Normal Space|normal space]]. | {{Recall|Normal Space|normal space|index = 2}}
{{:Definition:Normal Space/Definition 2}}
Let $T = \struct {S, \tau}$ be a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact]] [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
From [[Weakly Sigma-Locally Compact T2 Space is T4|Weakly $\sigma... | Weakly Sigma-Locally Compact T2 Space is Normal | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_T2_Space_is_Normal | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_T2_Space_is_Normal | [
"Weakly Sigma-Locally Compact Spaces",
"Normal Spaces",
"Hausdorff Spaces",
"Sequence of Implications of Compactness Properties in Hausdorff Space"
] | [
"Definition:T2 Space",
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:Normal Space"
] | [
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:T2 Space",
"Weakly Sigma-Locally Compact T2 Space is T4",
"Definition:T4 Space"
] |
proofwiki-23369 | Weakly Sigma-Locally Compact T2 Space is T4 | Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space which is weakly $\sigma$-locally compact.
Then $T$ is a $T_4$ space. | {{Recall|T4 Space|$T_4$ space}}
{{:Definition:T4 Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a weakly $\sigma$-locally compact $T_2$ (Hausdorff) space.
{{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
{{:Definition:Weakly Sigma-Locally Compact Space}}
{{ProofWanted}} | Let $T = \struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] which is [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact]].
Then $T$ is a [[Definition:T4 Space|$T_4$ space]]. | {{Recall|T4 Space|$T_4$ space}}
{{:Definition:T4 Space/Definition 1}}
Let $T = \struct {S, \tau}$ be a [[Definition:Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact]] [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
{{Recall|Weakly Sigma-Locally Compact Space|weakly $\sigma$-locally compact space}}
... | Weakly Sigma-Locally Compact T2 Space is T4 | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_T2_Space_is_T4 | https://proofwiki.org/wiki/Weakly_Sigma-Locally_Compact_T2_Space_is_T4 | [
"Weakly Sigma-Locally Compact Spaces",
"T4 Spaces",
"Hausdorff Spaces",
"Sequence of Implications of Compactness Properties in Hausdorff Space"
] | [
"Definition:T2 Space",
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:T4 Space"
] | [
"Definition:Weakly Sigma-Locally Compact Space",
"Definition:T2 Space"
] |
proofwiki-23370 | Weakly Locally Compact T2 Space is T3.5 | Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space which is weakly locally compact.
Then $T$ is a $T_{3 \frac 1 2}$ space. | Let $T = \struct {S, \tau}$ be a weakly locally compact $T_2$ space.
We have:
:$T_2$ Space is $R_1$
The result follows from Weakly Locally Compact $R_1$ Space is $T_{3 \frac 1 2}$.
{{qed}}
Category:Weakly Locally Compact Spaces
Category:T3.5 Spaces
Category:Hausdorff Spaces
Category:Sequence of Implications of Compactn... | Let $T = \struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] which is [[Definition:Weakly Locally Compact Space|weakly locally compact]].
Then $T$ is a [[Definition:T3.5 Space|$T_{3 \frac 1 2}$ space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Weakly Locally Compact Space|weakly locally compact]] [[Definition:T2 Space|$T_2$ space]].
We have:
:[[T2 Space is R1|$T_2$ Space is $R_1$]]
The result follows from [[Weakly Locally Compact R1 Space is T3.5|Weakly Locally Compact $R_1$ Space is $T_{3 \frac 1 2}$]].
{{qed... | Weakly Locally Compact T2 Space is T3.5 | https://proofwiki.org/wiki/Weakly_Locally_Compact_T2_Space_is_T3.5 | https://proofwiki.org/wiki/Weakly_Locally_Compact_T2_Space_is_T3.5 | [
"Weakly Locally Compact Spaces",
"T3.5 Spaces",
"Hausdorff Spaces",
"Sequence of Implications of Compactness Properties in Hausdorff Space"
] | [
"Definition:T2 Space",
"Definition:Weakly Locally Compact Space",
"Definition:T3.5 Space"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:T2 Space",
"T2 Space is R1",
"Weakly Locally Compact R1 Space is T3.5",
"Category:Weakly Locally Compact Spaces",
"Category:T3.5 Spaces",
"Category:Hausdorff Spaces",
"Category:Sequence of Implications of Compactness Properties in Hausdorff Space"... |
proofwiki-23371 | Weakly Locally Compact T2 Space is Completely Regular | Let $T = \struct {S, \tau}$ be a $T_2$ (Hausdorff) space which is weakly locally compact.
Then $T$ is a completely regular space. | {{Recall|Completely Regular Space|completely regular space|index = 3}}
{{:Definition:Completely Regular Space/Definition 3}}
Let $T = \struct {S, \tau}$ be a weakly locally compact $T_2$ (Hausdorff) space.
From Weakly Locally Compact $T_2$ Space is $T_{3 \frac 1 2}$, $T$ is a $T_{3 \frac 1 2}$ space.
Hence the result.
... | Let $T = \struct {S, \tau}$ be a [[Definition:T2 Space|$T_2$ (Hausdorff) space]] which is [[Definition:Weakly Locally Compact Space|weakly locally compact]].
Then $T$ is a [[Definition:Completely Regular Space|completely regular space]]. | {{Recall|Completely Regular Space|completely regular space|index = 3}}
{{:Definition:Completely Regular Space/Definition 3}}
Let $T = \struct {S, \tau}$ be a [[Definition:Weakly Locally Compact Space|weakly locally compact]] [[Definition:T2 Space|$T_2$ (Hausdorff) space]].
From [[Weakly Locally Compact T2 Space is T3... | Weakly Locally Compact T2 Space is Completely Regular | https://proofwiki.org/wiki/Weakly_Locally_Compact_T2_Space_is_Completely_Regular | https://proofwiki.org/wiki/Weakly_Locally_Compact_T2_Space_is_Completely_Regular | [
"Weakly Locally Compact Spaces",
"Completely Regular Spaces",
"Hausdorff Spaces",
"Sequence of Implications of Compactness Properties in Hausdorff Space"
] | [
"Definition:T2 Space",
"Definition:Weakly Locally Compact Space",
"Definition:Completely Regular Space"
] | [
"Definition:Weakly Locally Compact Space",
"Definition:T2 Space",
"Weakly Locally Compact T2 Space is T3.5",
"Definition:T3.5 Space"
] |
proofwiki-23372 | Second-Countable T3 Space is Lindelöf and T5 | Let $T = \struct {S, \tau}$ be a second-countable $T_3$ space.
Then $T$ is both a Lindelöf space and a $T_5$ space. | Let $T = \struct {S, \tau}$ be a second-countable $T_3$ space.
From Second-Countable Space is Lindelöf, $T$ is a Lindelöf space.
From Second-Countable $T_3$ Space is $T_5$, $T$ is a $T_5$ space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Second-Countable Space|second-countable]] [[Definition:T3 Space|$T_3$ space]].
Then $T$ is both a [[Definition:Lindelöf Space|Lindelöf space]] and a [[Definition:T5 Space|$T_5$ space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Second-Countable Space|second-countable]] [[Definition:T3 Space|$T_3$ space]].
From [[Second-Countable Space is Lindelöf]], $T$ is a [[Definition:Lindelöf Space|Lindelöf space]].
From [[Second-Countable T3 Space is T5|Second-Countable $T_3$ Space is $T_5$]], $T$ is a [[De... | Second-Countable T3 Space is Lindelöf and T5 | https://proofwiki.org/wiki/Second-Countable_T3_Space_is_Lindelöf_and_T5 | https://proofwiki.org/wiki/Second-Countable_T3_Space_is_Lindelöf_and_T5 | [
"T3 Spaces",
"Second-Countable Spaces",
"Lindelöf Spaces",
"T5 Spaces",
"Sequence of Implications of Compactness Properties in T3 Space"
] | [
"Definition:Second-Countable Space",
"Definition:T3 Space",
"Definition:Lindelöf Space",
"Definition:T5 Space"
] | [
"Definition:Second-Countable Space",
"Definition:T3 Space",
"Second-Countable Space is Lindelöf",
"Definition:Lindelöf Space",
"Second-Countable T3 Space is T5",
"Definition:T5 Space"
] |
proofwiki-23373 | Product of Non-Meager Spaces is not necessarily Non-Meager | Let $T_1$ and $T_2$ be non-meager topological spaces.
Let $T = T_1 \times T_2$ be the product space of $T_1$ and $T_2$.
Then it is not necessarily the case that $T$ is also a non-meager space. | {{ProofWanted|There will probably be a counterexample that can be found if we search hard enough}} | Let $T_1$ and $T_2$ be [[Definition:Non-Meager Space|non-meager topological spaces]].
Let $T = T_1 \times T_2$ be the [[Definition:Product Space (Topology)|product space]] of $T_1$ and $T_2$.
Then it is not necessarily the case that $T$ is also a [[Definition:Non-Meager Space|non-meager space]]. | {{ProofWanted|There will probably be a counterexample that can be found if we search hard enough}} | Product of Non-Meager Spaces is not necessarily Non-Meager | https://proofwiki.org/wiki/Product_of_Non-Meager_Spaces_is_not_necessarily_Non-Meager | https://proofwiki.org/wiki/Product_of_Non-Meager_Spaces_is_not_necessarily_Non-Meager | [
"Product Spaces",
"Non-Meager Spaces"
] | [
"Definition:Meager Space/Non-Meager",
"Definition:Product Space (Topology)",
"Definition:Meager Space/Non-Meager"
] | [] |
proofwiki-23374 | Sorgenfrey Line is Paracompact | The Sorgenfrey line is a paracompact space. | Let $T$ denote the Sorgenfrey line.
From Sorgenfrey Line satisfies all Separation Axioms, $T$ is a regular space.
Then we have:
:Sorgenfrey Line is Lindelöf
:Regular Lindelöf Space is Paracompact.
{{qed}} | The [[Definition:Sorgenfrey Line|Sorgenfrey line]] is a [[Definition:Paracompact Space|paracompact space]]. | Let $T$ denote the [[Definition:Sorgenfrey Line|Sorgenfrey line]].
From [[Sorgenfrey Line satisfies all Separation Axioms]], $T$ is a [[Definition:Regular Space|regular space]].
Then we have:
:[[Sorgenfrey Line is Lindelöf]]
:[[Regular Lindelöf Space is Paracompact]].
{{qed}} | Sorgenfrey Line is Paracompact | https://proofwiki.org/wiki/Sorgenfrey_Line_is_Paracompact | https://proofwiki.org/wiki/Sorgenfrey_Line_is_Paracompact | [
"Sorgenfrey Line",
"Examples of Paracompact Spaces"
] | [
"Definition:Sorgenfrey Line",
"Definition:Paracompact Space"
] | [
"Definition:Sorgenfrey Line",
"Sorgenfrey Line satisfies all Separation Axioms",
"Definition:Regular Space",
"Sorgenfrey Line is Lindelöf",
"Regular Lindelöf Space is Paracompact"
] |
proofwiki-23375 | Sorgenfrey's Half-Open Square Topology is Not Paracompact | Let $T$ be a Sorgenfrey's half-open square.
Then $T$ is not a paracompact space. | From Sorgenfrey's Half-Open Square Topology is Not Countably Paracompact, $T$ is not a countably paracompact space.
Hence from Paracompact Space is Countably Paracompact, $T$ cannot be a paracompact space.
{{qed}}
Category:Sorgenfrey's Half-Open Square Topology
Category:Examples of Paracompact Spaces
gm4ufdjt9hsiwyfmzy... | Let $T$ be a [[Definition:Sorgenfrey's Half-Open Square|Sorgenfrey's half-open square]].
Then $T$ is not a [[Definition:Paracompact Space|paracompact space]]. | From [[Sorgenfrey's Half-Open Square Topology is Not Countably Paracompact]], $T$ is not a [[Definition:Countably Paracompact Space|countably paracompact space]].
Hence from [[Paracompact Space is Countably Paracompact]], $T$ cannot be a [[Definition:Paracompact Space|paracompact space]].
{{qed}}
[[Category:Sorgenfre... | Sorgenfrey's Half-Open Square Topology is Not Paracompact | https://proofwiki.org/wiki/Sorgenfrey's_Half-Open_Square_Topology_is_Not_Paracompact | https://proofwiki.org/wiki/Sorgenfrey's_Half-Open_Square_Topology_is_Not_Paracompact | [
"Sorgenfrey's Half-Open Square Topology",
"Examples of Paracompact Spaces"
] | [
"Definition:Sorgenfrey's Half-Open Square Topology",
"Definition:Paracompact Space"
] | [
"Sorgenfrey's Half-Open Square Topology is Not Countably Paracompact",
"Definition:Countably Paracompact Space",
"Paracompact Space is Countably Paracompact",
"Definition:Paracompact Space",
"Category:Sorgenfrey's Half-Open Square Topology",
"Category:Examples of Paracompact Spaces"
] |
proofwiki-23376 | Sorgenfrey Line is Metacompact | The Sorgenfrey line is a metacompact space. | Follows from:
:Sorgenfrey Line is Paracompact
:Paracompact Space is Metacompact.
{{qed}}
Category:Sorgenfrey Line
Category:Examples of Metacompact Spaces
nv02tfv2k8bo2qfz64zxmb2jdl0qngi | The [[Definition:Sorgenfrey Line|Sorgenfrey line]] is a [[Definition:Metacompact Space|metacompact space]]. | Follows from:
:[[Sorgenfrey Line is Paracompact]]
:[[Paracompact Space is Metacompact]].
{{qed}}
[[Category:Sorgenfrey Line]]
[[Category:Examples of Metacompact Spaces]]
nv02tfv2k8bo2qfz64zxmb2jdl0qngi | Sorgenfrey Line is Metacompact | https://proofwiki.org/wiki/Sorgenfrey_Line_is_Metacompact | https://proofwiki.org/wiki/Sorgenfrey_Line_is_Metacompact | [
"Sorgenfrey Line",
"Examples of Metacompact Spaces",
"Sorgenfrey Line",
"Examples of Metacompact Spaces"
] | [
"Definition:Sorgenfrey Line",
"Definition:Metacompact Space"
] | [
"Sorgenfrey Line is Paracompact",
"Paracompact Space is Metacompact",
"Category:Sorgenfrey Line",
"Category:Examples of Metacompact Spaces"
] |
proofwiki-23377 | Sorgenfrey's Half-Open Square Topology is Not Metacompact | Let $T$ be a Sorgenfrey's half-open square.
Then $T$ is not a metacompact space. | {{ProofWanted}}
Category:Sorgenfrey's Half-Open Square Topology
Category:Examples of Metacompact Spaces
pxiwzunufnfkiuv0m7ekfysfaa9oa0a | Let $T$ be a [[Definition:Sorgenfrey's Half-Open Square|Sorgenfrey's half-open square]].
Then $T$ is not a [[Definition:Metacompact Space|metacompact space]]. | {{ProofWanted}}
[[Category:Sorgenfrey's Half-Open Square Topology]]
[[Category:Examples of Metacompact Spaces]]
pxiwzunufnfkiuv0m7ekfysfaa9oa0a | Sorgenfrey's Half-Open Square Topology is Not Metacompact | https://proofwiki.org/wiki/Sorgenfrey's_Half-Open_Square_Topology_is_Not_Metacompact | https://proofwiki.org/wiki/Sorgenfrey's_Half-Open_Square_Topology_is_Not_Metacompact | [
"Sorgenfrey's Half-Open Square Topology",
"Examples of Metacompact Spaces",
"Sorgenfrey's Half-Open Square Topology",
"Examples of Metacompact Spaces"
] | [
"Definition:Sorgenfrey's Half-Open Square Topology",
"Definition:Metacompact Space"
] | [
"Category:Sorgenfrey's Half-Open Square Topology",
"Category:Examples of Metacompact Spaces"
] |
proofwiki-23378 | Irreducible Space is not necessarily Ultraconnected | Let $T = \struct {S, \tau}$ be a topological space which is irreducible.
Then it is not necessarily the case that $T$ is ultraconnected. | Let $T = \struct {S, \tau_p}$ be a particular point space with at least three points.
From Particular Point Space is Irreducible, $T$ is an irreducible space.
From Particular Point Space is not Ultraconnected, $T$ is not an ultraconnected space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Irreducible Space|irreducible]].
Then it is not necessarily the case that $T$ is [[Definition:Ultraconnected Space|ultraconnected]]. | Let $T = \struct {S, \tau_p}$ be a [[Definition:Particular Point Topology|particular point space]] with at least three [[Definition:Point of Set|points]].
From [[Particular Point Space is Irreducible]], $T$ is an [[Definition:Irreducible Space|irreducible space]].
From [[Particular Point Space is not Ultraconnected]]... | Irreducible Space is not necessarily Ultraconnected | https://proofwiki.org/wiki/Irreducible_Space_is_not_necessarily_Ultraconnected | https://proofwiki.org/wiki/Irreducible_Space_is_not_necessarily_Ultraconnected | [
"Irreducible Spaces",
"Ultraconnected Spaces",
"Sequence of Implications of Connectedness Properties"
] | [
"Definition:Topological Space",
"Definition:Irreducible Space",
"Definition:Ultraconnected Space"
] | [
"Definition:Particular Point Topology",
"Definition:Element",
"Particular Point Space is Irreducible",
"Definition:Irreducible Space",
"Particular Point Space is not Ultraconnected",
"Definition:Ultraconnected Space"
] |
proofwiki-23379 | Ultraconnected Space is not necessarily Irreducible | Let $T = \struct {S, \tau}$ be a topological space which is ultraconnected.
Then it is not necessarily the case that $T$ is irreducible. | Let $T = \struct {S, \tau_{\bar p} }$ be an excluded point space with at least three points.
By Excluded Point Space is Ultraconnected, $T$ is an ultraconnected space.
By Excluded Point Space is not Irreducible, $T$ is not an irreducible space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Ultraconnected Space|ultraconnected]].
Then it is not necessarily the case that $T$ is [[Definition:Irreducible Space|irreducible]]. | Let $T = \struct {S, \tau_{\bar p} }$ be an [[Definition:Excluded Point Topology|excluded point space]] with at least three [[Definition:Point of Set|points]].
By [[Excluded Point Space is Ultraconnected]], $T$ is an [[Definition:Ultraconnected Space|ultraconnected space]].
By [[Excluded Point Space is not Irreducibl... | Ultraconnected Space is not necessarily Irreducible | https://proofwiki.org/wiki/Ultraconnected_Space_is_not_necessarily_Irreducible | https://proofwiki.org/wiki/Ultraconnected_Space_is_not_necessarily_Irreducible | [
"Ultraconnected Spaces",
"Irreducible Spaces",
"Sequence of Implications of Connectedness Properties"
] | [
"Definition:Topological Space",
"Definition:Ultraconnected Space",
"Definition:Irreducible Space"
] | [
"Definition:Excluded Point Topology",
"Definition:Element",
"Excluded Point Space is Ultraconnected",
"Definition:Ultraconnected Space",
"Excluded Point Space is not Irreducible",
"Definition:Irreducible Space"
] |
proofwiki-23380 | Components and Quasicomponents of Arens-Fort Space are Equal | Let $T = \struct {S, \tau}$ be the Arens-Fort space.
Let $A \subseteq S$ be a subset of $T$.
Then:
:$A$ is a component of $T$
{{iff}}:
:$A$ is a quasicomponent of $T$. | {{ProofWanted}}
Category:Arens-Fort Space
Category:Examples of Topological Components
Category:Examples of Quasicomponents
8txk5dn9z78hccegwsxeyv2p86v0fqp | Let $T = \struct {S, \tau}$ be the [[Definition:Arens-Fort Space|Arens-Fort space]].
Let $A \subseteq S$ be a [[Definition:Subset|subset]] of $T$.
Then:
:$A$ is a [[Definition:Component (Topology)|component]] of $T$
{{iff}}:
:$A$ is a [[Definition:Quasicomponent|quasicomponent]] of $T$. | {{ProofWanted}}
[[Category:Arens-Fort Space]]
[[Category:Examples of Topological Components]]
[[Category:Examples of Quasicomponents]]
8txk5dn9z78hccegwsxeyv2p86v0fqp | Components and Quasicomponents of Arens-Fort Space are Equal | https://proofwiki.org/wiki/Components_and_Quasicomponents_of_Arens-Fort_Space_are_Equal | https://proofwiki.org/wiki/Components_and_Quasicomponents_of_Arens-Fort_Space_are_Equal | [
"Arens-Fort Space",
"Examples of Topological Components",
"Examples of Quasicomponents"
] | [
"Definition:Arens-Fort Space",
"Definition:Subset",
"Definition:Component (Topology)",
"Definition:Quasicomponent"
] | [
"Category:Arens-Fort Space",
"Category:Examples of Topological Components",
"Category:Examples of Quasicomponents"
] |
proofwiki-23381 | Extended Topologist's Sine Curve is Path-Connected | Let $T$ be the extended topologist's sine curve.
Then $T$ is path-connected. | From Extended Topologist's Sine Curve is Injectively Path-Connected we have that $T$ is injectively path-connected.
The result follows from Injectively Path-Connected Space is Path-Connected.
{{qed}} | Let $T$ be the [[Definition:Extended Topologist's Sine Curve|extended topologist's sine curve]].
Then $T$ is [[Definition:Path-Connected Space|path-connected]]. | From [[Extended Topologist's Sine Curve is Injectively Path-Connected]] we have that $T$ is [[Definition:Injectively Path-Connected Space|injectively path-connected]].
The result follows from [[Injectively Path-Connected Space is Path-Connected]].
{{qed}} | Extended Topologist's Sine Curve is Path-Connected | https://proofwiki.org/wiki/Extended_Topologist's_Sine_Curve_is_Path-Connected | https://proofwiki.org/wiki/Extended_Topologist's_Sine_Curve_is_Path-Connected | [
"Extended Topologist's Sine Curve",
"Examples of Path-Connected Spaces"
] | [
"Definition:Extended Topologist's Sine Curve",
"Definition:Path-Connected/Topological Space"
] | [
"Extended Topologist's Sine Curve is Injectively Path-Connected",
"Definition:Injectively Path-Connected/Topological Space",
"Injectively Path-Connected Space is Path-Connected"
] |
proofwiki-23382 | Union of Adjacent Open Intervals is Locally Path-Connected | Let $T = \struct {S, \tau_S}$ be the union of adjacent open intervals:
:$S = \openint a b \cup \openint b c$
where:
:$a, b, c \in \R: a < b < c$.
:$\tau_S$ is the subspace topology induced on $S$ by the usual topology $\tau_d$ on the real numbers $\R$.
Then $T$ is locally path-connected. | From Union of Adjacent Open Intervals is Locally Injectively Path-Connected, $T$ is locally injectively path-connected.
The result follows from Locally Injectively Path-Connected Space is Locally Path-Connected.
{{qed}}
Category:Union of Adjacent Open Intervals
Category:Examples of Locally Path-Connected Spaces
h4vq2tt... | Let $T = \struct {S, \tau_S}$ be the [[Definition:Union of Adjacent Open Intervals|union of adjacent open intervals]]:
:$S = \openint a b \cup \openint b c$
where:
:$a, b, c \in \R: a < b < c$.
:$\tau_S$ is the [[Definition:Subspace Topology|subspace topology induced]] on $S$ by the [[Definition:Usual Topology|usual t... | From [[Union of Adjacent Open Intervals is Locally Injectively Path-Connected]], $T$ is [[Definition:Locally Injectively Path-Connected Space|locally injectively path-connected]].
The result follows from [[Locally Injectively Path-Connected Space is Locally Path-Connected]].
{{qed}}
[[Category:Union of Adjacent Open ... | Union of Adjacent Open Intervals is Locally Path-Connected | https://proofwiki.org/wiki/Union_of_Adjacent_Open_Intervals_is_Locally_Path-Connected | https://proofwiki.org/wiki/Union_of_Adjacent_Open_Intervals_is_Locally_Path-Connected | [
"Union of Adjacent Open Intervals",
"Examples of Locally Path-Connected Spaces"
] | [
"Definition:Union of Adjacent Open Intervals",
"Definition:Topological Subspace",
"Definition:Euclidean Space/Euclidean Topology",
"Definition:Real Number",
"Definition:Locally Path-Connected Space"
] | [
"Union of Adjacent Open Intervals is Locally Injectively Path-Connected",
"Definition:Locally Injectively Path-Connected Space",
"Locally Injectively Path-Connected Space is Locally Path-Connected",
"Category:Union of Adjacent Open Intervals",
"Category:Examples of Locally Path-Connected Spaces"
] |
proofwiki-23383 | Union of Adjacent Open Intervals is not Path-Connected | Let $T = \struct {S, \tau_S}$ be the union of adjacent open intervals:
:$S = \openint a b \cup \openint b c$
where:
:$a, b, c \in \R: a < b < c$.
:$\tau_S$ is the subspace topology induced on $S$ by the usual topology $\tau_d$ on the real numbers $\R$.
Then $T$ is ''not'' path-connected. | {{Recall|Path-Connected Space|path-connected space}}
{{:Definition:Path-Connected Space}}
So, let $T$ be the union of adjacent open intervals as defined {{hypothesis}}.
Let $x \in \openint a b$.
Let $y \in \openint b c$.
Let $\I := \closedint 0 1$ denote the closed unit interval.
{{AimForCont}} there exists a path $f: ... | Let $T = \struct {S, \tau_S}$ be the [[Definition:Union of Adjacent Open Intervals|union of adjacent open intervals]]:
:$S = \openint a b \cup \openint b c$
where:
:$a, b, c \in \R: a < b < c$.
:$\tau_S$ is the [[Definition:Subspace Topology|subspace topology induced]] on $S$ by the [[Definition:Usual Topology|usual t... | {{Recall|Path-Connected Space|path-connected space}}
{{:Definition:Path-Connected Space}}
So, let $T$ be the [[Definition:Union of Adjacent Open Intervals|union of adjacent open intervals]] as defined {{hypothesis}}.
Let $x \in \openint a b$.
Let $y \in \openint b c$.
Let $\I := \closedint 0 1$ denote the [[Defini... | Union of Adjacent Open Intervals is not Path-Connected | https://proofwiki.org/wiki/Union_of_Adjacent_Open_Intervals_is_not_Path-Connected | https://proofwiki.org/wiki/Union_of_Adjacent_Open_Intervals_is_not_Path-Connected | [
"Union of Adjacent Open Intervals",
"Examples of Path-Connected Spaces"
] | [
"Definition:Union of Adjacent Open Intervals",
"Definition:Topological Subspace",
"Definition:Euclidean Space/Euclidean Topology",
"Definition:Real Number",
"Definition:Path-Connected/Topological Space"
] | [
"Definition:Union of Adjacent Open Intervals",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Path (Topology)",
"Continuous Image of Closed Real Interval is Closed Real Interval",
"Definition:Union of Adjacent Open Intervals",
"Definition:Contradiction",
"Definition:Path-Connected/Topologi... |
proofwiki-23384 | Union of Adjacent Open Intervals is Locally Injectively Path-Connected | Let $T = \struct {S, \tau_S}$ be the union of adjacent open intervals:
:$S = \openint a b \cup \openint b c$
where:
:$a, b, c \in \R: a < b < c$.
:$\tau_S$ is the subspace topology induced on $S$ by the usual topology $\tau_d$ on the real numbers $\R$.
Then $T$ is locally injectively path-connected. | {{Recall|Locally Injectively Path-Connected Space|locally injectively path-connected space}}
{{Definition:Locally Injectively Path-Connected Space}}
Consider:
:$\BB := \set {\openint x y: x, y \in \R \cap \openint a b, x < y} \cup \set {\openint x y: x, y \in \R \cap \openint b c, x < y}$
From Basis for Union of Adjac... | Let $T = \struct {S, \tau_S}$ be the [[Definition:Union of Adjacent Open Intervals|union of adjacent open intervals]]:
:$S = \openint a b \cup \openint b c$
where:
:$a, b, c \in \R: a < b < c$.
:$\tau_S$ is the [[Definition:Subspace Topology|subspace topology induced]] on $S$ by the [[Definition:Usual Topology|usual t... | {{Recall|Locally Injectively Path-Connected Space|locally injectively path-connected space}}
{{Definition:Locally Injectively Path-Connected Space}}
Consider:
:$\BB := \set {\openint x y: x, y \in \R \cap \openint a b, x < y} \cup \set {\openint x y: x, y \in \R \cap \openint b c, x < y}$
From [[Basis for Union of A... | Union of Adjacent Open Intervals is Locally Injectively Path-Connected | https://proofwiki.org/wiki/Union_of_Adjacent_Open_Intervals_is_Locally_Injectively_Path-Connected | https://proofwiki.org/wiki/Union_of_Adjacent_Open_Intervals_is_Locally_Injectively_Path-Connected | [
"Union of Adjacent Open Intervals",
"Examples of Locally Injectively Path-Connected Spaces"
] | [
"Definition:Union of Adjacent Open Intervals",
"Definition:Topological Subspace",
"Definition:Euclidean Space/Euclidean Topology",
"Definition:Real Number",
"Definition:Locally Injectively Path-Connected Space"
] | [
"Basis for Union of Adjacent Open Intervals",
"Definition:Basis (Topology)",
"Real Interval is Injectively Path-Connected",
"Definition:Real Interval",
"Definition:Injectively Path-Connected/Subset",
"Category:Union of Adjacent Open Intervals",
"Category:Examples of Locally Injectively Path-Connected Sp... |
proofwiki-23385 | Union of Adjacent Open Intervals is not Injectively Path-Connected | Let $T = \struct {S, \tau_S}$ be the union of adjacent open intervals:
:$S = \openint a b \cup \openint b c$
where:
:$a, b, c \in \R: a < b < c$.
:$\tau_S$ is the subspace topology induced on $S$ by the usual topology $\tau_d$ on the real numbers $\R$.
Then $T$ is ''not'' injectively path-connected. | From Union of Adjacent Open Intervals is not Path-Connected, $T$ is not path-connected.
The result follows from Injectively Path-Connected Space is Path-Connected.
{{qed}}
Category:Union of Adjacent Open Intervals
Category:Examples of Injectively Path-Connected Spaces
rxq8yynk8544lhbkad118ybna902rs5 | Let $T = \struct {S, \tau_S}$ be the [[Definition:Union of Adjacent Open Intervals|union of adjacent open intervals]]:
:$S = \openint a b \cup \openint b c$
where:
:$a, b, c \in \R: a < b < c$.
:$\tau_S$ is the [[Definition:Subspace Topology|subspace topology induced]] on $S$ by the [[Definition:Usual Topology|usual t... | From [[Union of Adjacent Open Intervals is not Path-Connected]], $T$ is not [[Definition:Path-Connected Space|path-connected]].
The result follows from [[Injectively Path-Connected Space is Path-Connected]].
{{qed}}
[[Category:Union of Adjacent Open Intervals]]
[[Category:Examples of Injectively Path-Connected Spaces... | Union of Adjacent Open Intervals is not Injectively Path-Connected | https://proofwiki.org/wiki/Union_of_Adjacent_Open_Intervals_is_not_Injectively_Path-Connected | https://proofwiki.org/wiki/Union_of_Adjacent_Open_Intervals_is_not_Injectively_Path-Connected | [
"Union of Adjacent Open Intervals",
"Examples of Injectively Path-Connected Spaces"
] | [
"Definition:Union of Adjacent Open Intervals",
"Definition:Topological Subspace",
"Definition:Euclidean Space/Euclidean Topology",
"Definition:Real Number",
"Definition:Injectively Path-Connected/Topological Space"
] | [
"Union of Adjacent Open Intervals is not Path-Connected",
"Definition:Path-Connected/Topological Space",
"Injectively Path-Connected Space is Path-Connected",
"Category:Union of Adjacent Open Intervals",
"Category:Examples of Injectively Path-Connected Spaces"
] |
proofwiki-23386 | Real Interval is Injectively Path-Connected | Let $\struct {\R, \tau_d}$ be the real number line under the Euclidean topology.
Let $T = \struct {S, \tau}$ be a real interval under the topology induced by $\tau_d$, which is also the Euclidean topology.
Then $T$ is injectively path-connected. | {{Recall|Injectively Path-Connected Space|injectively path-connected space}}
{{Definition:Injectively Path-Connected Space}}
Let $a, b \in S$ be arbitrary such that $a < b$.
Let $\I$ denote the closed unit interval $\closedint 0 1$.
Let us define the injection $f: \I \to S$ as:
:$\forall x \in \I: \map f x = a + \paren... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line]] under the [[Definition:Euclidean Topology|Euclidean topology]].
Let $T = \struct {S, \tau}$ be a [[Definition:Real Interval|real interval]] under the [[Definition:Induced Topology|topology induced]] by $\tau_d$, ... | {{Recall|Injectively Path-Connected Space|injectively path-connected space}}
{{Definition:Injectively Path-Connected Space}}
Let $a, b \in S$ be [[Definition:Arbitrary|arbitrary]] such that $a < b$.
Let $\I$ denote the [[Definition:Closed Unit Interval|closed unit interval]] $\closedint 0 1$.
Let us define the [[Def... | Real Interval is Injectively Path-Connected | https://proofwiki.org/wiki/Real_Interval_is_Injectively_Path-Connected | https://proofwiki.org/wiki/Real_Interval_is_Injectively_Path-Connected | [
"Real Intervals",
"Examples of Injectively Path-Connected Spaces"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology",
"Definition:Real Interval",
"Definition:Induced Topology",
"Definition:Euclidean Space/Euclidean Topology",
"Definition:Injectively Path-Connected/Topological Space"
] | [
"Definition:Arbitrary",
"Definition:Real Interval/Unit Interval/Closed",
"Definition:Injection",
"Definition:Injection",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Arbitrary",
"Universal Generalisation",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Injec... |
proofwiki-23387 | Basis for Union of Adjacent Open Intervals | <onlyinclude>
Let $T = \struct {S, \tau_S}$ be the union of adjacent open intervals considered as a topological space:
:$S = \openint a b \cup \openint b c$
Let $\BB_1$ and $\BB_2$ be the set of subsets of $\R$ defined as:
{{begin-eqn}}
{{eqn | l = \BB_1
| r = \set {\openint x y: x, y \in \R \cap \openint a b, x... | For $x \in S$, let:
:$\VV = \set {\openint s t: a < s < x < t < b} \cup \set {\openint s t: c < s < x < t < d}$
From Local Basis for Point in Union of Adjacent Open Intervals, $\VV$ is a local basis for $x$ in $T$.
Let $\openint s t \in \bigcup \VV$.
Then either:
:$\openint s t \in \BB_1$
or:
:$\openint s t \in \BB_2$
... | <onlyinclude>
Let $T = \struct {S, \tau_S}$ be the [[Definition:Union of Adjacent Open Intervals|union of adjacent open intervals]] considered as a [[Definition:Topological Space|topological space]]:
:$S = \openint a b \cup \openint b c$
Let $\BB_1$ and $\BB_2$ be the [[Definition:Set of Sets|set of subsets]] of $\R... | For $x \in S$, let:
:$\VV = \set {\openint s t: a < s < x < t < b} \cup \set {\openint s t: c < s < x < t < d}$
From [[Local Basis for Point in Union of Adjacent Open Intervals]], $\VV$ is a [[Definition:Local Basis|local basis]] for $x$ in $T$.
Let $\openint s t \in \bigcup \VV$.
Then either:
:$\openint s t \in \BB... | Basis for Union of Adjacent Open Intervals | https://proofwiki.org/wiki/Basis_for_Union_of_Adjacent_Open_Intervals | https://proofwiki.org/wiki/Basis_for_Union_of_Adjacent_Open_Intervals | [
"Union of Adjacent Open Intervals",
"Examples of Topological Bases"
] | [
"Definition:Union of Adjacent Open Intervals",
"Definition:Topological Space",
"Definition:Set of Sets",
"Definition:Basis (Topology)"
] | [
"Local Basis for Point in Union of Adjacent Open Intervals",
"Definition:Local Basis",
"Definition:Set Union",
"Definition:Set Equality",
"Union of Local Bases is Basis",
"Definition:Basis (Topology)",
"Category:Union of Adjacent Open Intervals",
"Category:Examples of Topological Bases"
] |
proofwiki-23388 | Local Basis for Point on Real Number Line | Let $\struct {\R, \tau_d}$ be the real number line under the Euclidean topology.
Let $x \in \R$ be arbitrary.
Let $\BB = \set {\openint a b: a, b \in \R: a < x < b}$
Then $\BB$ is a local basis for $x$ in $\R$. | Let $x \in \R$ be arbitrary.
{{Recall|Local Basis|local basis}}
{{:Definition:Local Basis/Local Basis for Open Sets}}
So, let $U = \openint a b$ be an arbitrary element of $\tau_d$ such that $x \in U$.
We have trivially that $\openint a b \in \BB$ by definition.
Hence $\BB$ is a local basis for $x$ in $\R$.
{{qed}}
Cat... | Let $\struct {\R, \tau_d}$ be the [[Definition:Real Number Line with Euclidean Topology|real number line]] under the [[Definition:Euclidean Topology|Euclidean topology]].
Let $x \in \R$ be [[Definition:Arbitrary|arbitrary]].
Let $\BB = \set {\openint a b: a, b \in \R: a < x < b}$
Then $\BB$ is a [[Definition:Local B... | Let $x \in \R$ be [[Definition:Arbitrary|arbitrary]].
{{Recall|Local Basis|local basis}}
{{:Definition:Local Basis/Local Basis for Open Sets}}
So, let $U = \openint a b$ be an [[Definition:Arbitrary|arbitrary]] [[Definition:Element|element]] of $\tau_d$ such that $x \in U$.
We have trivially that $\openint a b \in \... | Local Basis for Point on Real Number Line | https://proofwiki.org/wiki/Local_Basis_for_Point_on_Real_Number_Line | https://proofwiki.org/wiki/Local_Basis_for_Point_on_Real_Number_Line | [
"Real Number Line with Euclidean Topology",
"Examples of Local Bases"
] | [
"Definition:Euclidean Space/Euclidean Topology/Real Number Line",
"Definition:Euclidean Space/Euclidean Topology",
"Definition:Arbitrary",
"Definition:Local Basis"
] | [
"Definition:Arbitrary",
"Definition:Arbitrary",
"Definition:Element",
"Definition:Local Basis",
"Category:Real Number Line with Euclidean Topology",
"Category:Examples of Local Bases"
] |
proofwiki-23389 | Preimage of Separation under Continuous Mapping is Separation | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be topological spaces.
Let $f: T_1 \to T_2$ be a continuous mapping.
Let $A, B \subseteq S_2$ such that $A \mid B$.
That is, such that $A$ and $B$ separate $T_2$.
Then their preimages $f^{-1} \sqbrk A$ and $f^{-1} \sqbrk B$ form a separation of $f^{-1}... | From Components of Separation are Clopen, we have that $A$ and $B$ are both clopen.
Hence from Clopen Set and Complement form Separation:
:$B = \relcomp {S_2} A$
and:
:$A = \relcomp {S_2} B$
From Complement of Preimage equals Preimage of Complement:
:$\relcomp {S_1} {f^{-1} \sqbrk A} = f^{-1} \sqbrk {\relcomp {S_2} A}$... | Let $T_1 = \struct {S_1, \tau_1}$ and $T_2 = \struct {S_2, \tau_2}$ be [[Definition:Topological Space|topological spaces]].
Let $f: T_1 \to T_2$ be a [[Definition:Everywhere Continuous Mapping (Topology)|continuous mapping]].
Let $A, B \subseteq S_2$ such that $A \mid B$.
That is, such that $A$ and $B$ [[Definition:... | From [[Components of Separation are Clopen]], we have that $A$ and $B$ are both [[Definition:Clopen Set|clopen]].
Hence from [[Clopen Set and Complement form Separation]]:
:$B = \relcomp {S_2} A$
and:
:$A = \relcomp {S_2} B$
From [[Complement of Preimage equals Preimage of Complement]]:
:$\relcomp {S_1} {f^{-1} \sqb... | Preimage of Separation under Continuous Mapping is Separation | https://proofwiki.org/wiki/Preimage_of_Separation_under_Continuous_Mapping_is_Separation | https://proofwiki.org/wiki/Preimage_of_Separation_under_Continuous_Mapping_is_Separation | [
"Continuous Mappings (Topology)",
"Separations"
] | [
"Definition:Topological Space",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Definition:Separation (Topology)",
"Definition:Preimage/Mapping/Subset",
"Definition:Separation (Topology)"
] | [
"Components of Separation are Clopen",
"Definition:Clopen Set",
"Clopen Set and Complement form Separation",
"Complement of Preimage equals Preimage of Complement",
"Definition:Disjoint Sets",
"Definition:Separation (Topology)",
"Definition:Contradiction",
"Definition:Separation (Topology)",
"Catego... |
proofwiki-23390 | Totally Separated Space is Urysohn | Let $T = \struct {S, \tau}$ be a topological space which is totally separated.
Then $T$ is an Urysohn space. | {{Recall|Urysohn Space|Urysohn space}}
{{:Definition:Urysohn Space}}
Let $T = \struct {S, \tau}$ be a totally separated space.
{{Recall|Totally Separated Space|totally separated space}}
{{:Definition:Totally Separated Space/Definition 1}}
Consider a mapping $f: S \to \closedint 0 1$ such that:
{{begin-eqn}}
{{eqn | l =... | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Totally Separated Space|totally separated]].
Then $T$ is an [[Definition:Urysohn Space|Urysohn space]]. | {{Recall|Urysohn Space|Urysohn space}}
{{:Definition:Urysohn Space}}
Let $T = \struct {S, \tau}$ be a [[Definition:Totally Separated Space|totally separated space]].
{{Recall|Totally Separated Space|totally separated space}}
{{:Definition:Totally Separated Space/Definition 1}}
Consider a [[Definition:Mapping|mapping... | Totally Separated Space is Urysohn | https://proofwiki.org/wiki/Totally_Separated_Space_is_Urysohn | https://proofwiki.org/wiki/Totally_Separated_Space_is_Urysohn | [
"Totally Separated Spaces",
"Urysohn Spaces",
"Sequence of Implications of Disconnectedness Properties"
] | [
"Definition:Topological Space",
"Definition:Totally Separated Space",
"Definition:Urysohn Space"
] | [
"Definition:Totally Separated Space",
"Definition:Mapping",
"Definition:Restriction/Mapping",
"Definition:Image (Set Theory)/Mapping/Subset",
"Definition:Mapping",
"Constant Mapping is Continuous",
"Definition:Continuous Mapping (Topology)/Everywhere",
"Continuous Mapping of Separation",
"Definition... |
proofwiki-23391 | Totally Separated Space is T2.5 | Let $T = \struct {S, \tau}$ be a topological space which is totally separated.
Then $T$ is a $T_{2 \frac 1 2}$ space. | Let $T = \struct {S, \tau}$ be a totally separated space.
From Totally Separated Space is Urysohn, $T$ is an Urysohn space.
It follows from Urysohn Space is $T_{2 \frac 1 2}$ Space that $T$ is also a $T_{2 \frac 1 2}$ space.
{{qed}} | Let $T = \struct {S, \tau}$ be a [[Definition:Topological Space|topological space]] which is [[Definition:Totally Separated Space|totally separated]].
Then $T$ is a [[Definition:T2.5 Space|$T_{2 \frac 1 2}$ space]]. | Let $T = \struct {S, \tau}$ be a [[Definition:Totally Separated Space|totally separated space]].
From [[Totally Separated Space is Urysohn]], $T$ is an [[Definition:Urysohn Space|Urysohn space]].
It follows from [[Urysohn Space is T2.5|Urysohn Space is $T_{2 \frac 1 2}$ Space]] that $T$ is also a [[Definition:T2.5 Sp... | Totally Separated Space is T2.5 | https://proofwiki.org/wiki/Totally_Separated_Space_is_T2.5 | https://proofwiki.org/wiki/Totally_Separated_Space_is_T2.5 | [
"Totally Separated Spaces",
"T2.5 Spaces"
] | [
"Definition:Topological Space",
"Definition:Totally Separated Space",
"Definition:T2.5 Space"
] | [
"Definition:Totally Separated Space",
"Totally Separated Space is Urysohn",
"Definition:Urysohn Space",
"Urysohn Space is T2.5",
"Definition:T2.5 Space"
] |
proofwiki-23392 | Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products/Necessary Condition | Let $\mathbf C$ be a locally small category.
Let $\mathbf C \times \mathbf C$ be the product category of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the diagonal functor.
Let $\Delta$ have a right adjoint.
Then $\mathbf C$ has all binary products such that the right adjoint is... | Let $\otimes:\mathbf {C \times C} \to \mathbf C$ be a right adjoint of $\Delta_{\mathbf J}$.
==== $\mathbf C$ has all Binary Products ====
By definition of right adjoint there exists an adjunction $\tuple{\Delta, \otimes, \alpha}$.
Let $\pi : \Delta \otimes \to \operatorname{id}_{\mathbf {C \times C} }$ denote the coun... | Let $\mathbf C$ be a [[Definition:Locally Small Category|locally small category]].
Let $\mathbf C \times \mathbf C$ be the [[Definition:Product Category|product category]] of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the [[Definition:Diagonal Functor on Product Category|di... | Let $\otimes:\mathbf {C \times C} \to \mathbf C$ be a [[Definition:Right Adjoint Functor|right adjoint]] of $\Delta_{\mathbf J}$.
==== $\mathbf C$ has all Binary Products ====
By definition of [[Definition:Right Adjoint Functor|right adjoint]] there exists an [[Definition:Adjunction|adjunction]] $\tuple{\Delta, \otime... | Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products/Necessary Condition | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Right_Adjoint_Iff_Category_has_Binary_Products/Necessary_Condition | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Right_Adjoint_Iff_Category_has_Binary_Products/Necessary_Condition | [
"Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products"
] | [
"Definition:Locally Small Category",
"Definition:Product Category",
"Definition:Diagonal Functor/Product Category",
"Definition:Right Adjoint Functor",
"Definition:Product (Category Theory)/Binary Product",
"Definition:Right Adjoint Functor",
"Definition:Product Functor"
] | [
"Definition:Right Adjoint Functor",
"Definition:Right Adjoint Functor",
"Definition:Adjunction",
"Definition:Counit of Adjunction",
"Morphism of Counit of Adjunction is Universal",
"Definition:Object (Category Theory)",
"Definition:Morphism",
"Definition:Universal Morphism from Functor to Object",
"... |
proofwiki-23393 | Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products/Sufficient Condition | Let $\mathbf C$ be a locally small category.
Let $\mathbf C \times \mathbf C$ be the product category of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the diagonal functor.
Let $\mathbf C$ have all binary products.
Then $\Delta$ has a right adjoint. | Let $\times: \mathbf {C \times C} \to \mathbf C$ denote the product functor.
It will be shown that there exists an adjunction $\tuple {\Delta, \times, \alpha}$.
==== Construction of $\alpha$ ====
We construct $\alpha$ as follows.
Let $X \in \mathbf C$ and $\tuple{C_1, C_2} \in \mathbf{C \times C}$ be objects.
Let $\tup... | Let $\mathbf C$ be a [[Definition:Locally Small Category|locally small category]].
Let $\mathbf C \times \mathbf C$ be the [[Definition:Product Category|product category]] of $\mathbf C$ with itself.
Let $\Delta: \mathbf C \to \mathbf C \times \mathbf C$ denote the [[Definition:Diagonal Functor on Product Category|di... | Let $\times: \mathbf {C \times C} \to \mathbf C$ denote the [[Definition:Product Functor|product functor]].
It will be shown that there exists an [[Definition:Adjunction|adjunction]] $\tuple {\Delta, \times, \alpha}$.
==== Construction of $\alpha$ ====
We construct $\alpha$ as follows.
Let $X \in \mathbf C$ and ... | Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products/Sufficient Condition | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Right_Adjoint_Iff_Category_has_Binary_Products/Sufficient_Condition | https://proofwiki.org/wiki/Diagonal_Functor_on_Product_Category_has_Right_Adjoint_Iff_Category_has_Binary_Products/Sufficient_Condition | [
"Diagonal Functor on Product Category has Right Adjoint Iff Category has Binary Products"
] | [
"Definition:Locally Small Category",
"Definition:Product Category",
"Definition:Diagonal Functor/Product Category",
"Definition:Product (Category Theory)/Binary Product",
"Definition:Right Adjoint Functor"
] | [
"Definition:Product Functor",
"Definition:Adjunction",
"Definition:Object (Category Theory)",
"Definition:Product (Category Theory)/Binary Product",
"Definition:Product (Category Theory)/Binary Product Projection",
"Definition:Product (Category Theory)/Binary Product",
"Definition:Ordered Tuple as Order... |
proofwiki-23394 | Divisor Space is Connected | Let $T = \struct {S, \tau}$ be the divisor space.
Then $T$ is connected. | {{ProofWanted}}
Category:Divisor Topology
Category:Examples of Connected Topological Spaces
7yu2oq8xxzyg2xr10xjotqcxws3jm7s | Let $T = \struct {S, \tau}$ be the [[Definition:Divisor Space|divisor space]].
Then $T$ is [[Definition:Connected Topological Space|connected]]. | {{ProofWanted}}
[[Category:Divisor Topology]]
[[Category:Examples of Connected Topological Spaces]]
7yu2oq8xxzyg2xr10xjotqcxws3jm7s | Divisor Space is Connected | https://proofwiki.org/wiki/Divisor_Space_is_Connected | https://proofwiki.org/wiki/Divisor_Space_is_Connected | [
"Divisor Topology",
"Examples of Connected Topological Spaces"
] | [
"Definition:Divisor Topology",
"Definition:Connected Topological Space"
] | [
"Category:Divisor Topology",
"Category:Examples of Connected Topological Spaces"
] |
proofwiki-23395 | Divisor Space is not T1 | Let $T = \struct {S, \tau}$ be the divisor space.
Then $T$ is not a $T_1$ space. | {{ProofWanted}}
Category:Divisor Topology
Category:Examples of T1 Spaces
dgfpuzigexqbzwdoctmtbclbo7axl1r | Let $T = \struct {S, \tau}$ be the [[Definition:Divisor Space|divisor space]].
Then $T$ is not a [[Definition:T1 Space|$T_1$ space]]. | {{ProofWanted}}
[[Category:Divisor Topology]]
[[Category:Examples of T1 Spaces]]
dgfpuzigexqbzwdoctmtbclbo7axl1r | Divisor Space is not T1 | https://proofwiki.org/wiki/Divisor_Space_is_not_T1 | https://proofwiki.org/wiki/Divisor_Space_is_not_T1 | [
"Divisor Topology",
"Examples of T1 Spaces",
"Divisor Topology",
"Examples of T1 Spaces"
] | [
"Definition:Divisor Topology",
"Definition:T1 Space"
] | [
"Category:Divisor Topology",
"Category:Examples of T1 Spaces"
] |
proofwiki-23396 | Divisor Space is Scattered | Let $T = \struct {S, \tau}$ be the divisor space.
Then $T$ is a scattered space. | {{ProofWanted}}
Category:Divisor Topology
Category:Examples of Scattered Spaces
ifwqtjd2azeup6wtpg4iswwt3ce63vi | Let $T = \struct {S, \tau}$ be the [[Definition:Divisor Space|divisor space]].
Then $T$ is a [[Definition:Scattered Space|scattered space]]. | {{ProofWanted}}
[[Category:Divisor Topology]]
[[Category:Examples of Scattered Spaces]]
ifwqtjd2azeup6wtpg4iswwt3ce63vi | Divisor Space is Scattered | https://proofwiki.org/wiki/Divisor_Space_is_Scattered | https://proofwiki.org/wiki/Divisor_Space_is_Scattered | [
"Divisor Topology",
"Examples of Scattered Spaces",
"Divisor Topology",
"Examples of Scattered Spaces"
] | [
"Definition:Divisor Topology",
"Definition:Scattered Space"
] | [
"Category:Divisor Topology",
"Category:Examples of Scattered Spaces"
] |
proofwiki-23397 | Power Series Expansion for Sinc Function | {{begin-eqn}}
{{eqn | l = \map \sinc x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } { \paren {2 n + 1}!}
| c =
}}
{{eqn | r = 1 - \dfrac {x^2} { 3!} + \dfrac {x^4} { 5!} - \dfrac {x^6} { 7!} + \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map \sinc x
| r = \frac {\map \sin x} x
| c = {{Defof|Sinc Function}}
}}
{{eqn | r = \frac 1 x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}
| c = Power Series Expansion for Sine Function
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \paren ... | {{begin-eqn}}
{{eqn | l = \map \sinc x
| r = \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n} } { \paren {2 n + 1}!}
| c =
}}
{{eqn | r = 1 - \dfrac {x^2} { 3!} + \dfrac {x^4} { 5!} - \dfrac {x^6} { 7!} + \cdots
| c =
}}
{{end-eqn}} | {{begin-eqn}}
{{eqn | l = \map \sinc x
| r = \frac {\map \sin x} x
| c = {{Defof|Sinc Function}}
}}
{{eqn | r = \frac 1 x \sum_{n \mathop = 0}^\infty \paren {-1}^n \frac {x^{2 n + 1} } {\paren {2 n + 1}!}
| c = [[Power Series Expansion for Sine Function]]
}}
{{eqn | r = \sum_{n \mathop = 0}^\infty \pa... | Power Series Expansion for Sinc Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Sinc_Function | https://proofwiki.org/wiki/Power_Series_Expansion_for_Sinc_Function | [
"Sinc Function",
"Examples of Power Series"
] | [] | [
"Power Series Expansion for Sine Function",
"Category:Sinc Function",
"Category:Examples of Power Series"
] |
proofwiki-23398 | Derivative of Sinc Function | :$\dfrac \d {\d x} \paren {\map \sinc x} = \dfrac {x \cos x - \sin x} {x^2}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\dfrac {\sin x} x}
| r = \dfrac {x \map {\frac \d {\d x} } {\sin x} - \sin x \map {\frac \d {\d x} } x} {x^2}
| c = Quotient Rule for Derivatives
}}
{{eqn | r = \dfrac {x \cdot \cos x - \sin x \cdot 1} {x^2}
| c = Derivative of Sine Function, Power R... | :$\dfrac \d {\d x} \paren {\map \sinc x} = \dfrac {x \cos x - \sin x} {x^2}$ | {{begin-eqn}}
{{eqn | l = \map {\dfrac \d {\d x} } {\dfrac {\sin x} x}
| r = \dfrac {x \map {\frac \d {\d x} } {\sin x} - \sin x \map {\frac \d {\d x} } x} {x^2}
| c = [[Quotient Rule for Derivatives]]
}}
{{eqn | r = \dfrac {x \cdot \cos x - \sin x \cdot 1} {x^2}
| c = [[Derivative of Sine Function]],... | Derivative of Sinc Function | https://proofwiki.org/wiki/Derivative_of_Sinc_Function | https://proofwiki.org/wiki/Derivative_of_Sinc_Function | [
"Sinc Function",
"Derivatives"
] | [] | [
"Quotient Rule for Derivatives",
"Derivative of Sine Function",
"Power Rule for Derivatives",
"Category:Sinc Function",
"Category:Derivatives"
] |
proofwiki-23399 | Primitive of Sinc Function | :$\ds \int_{t \mathop \to 0}^{t \mathop = x} \frac {\sin t} t \rd t = \map \Si x $ | True by {{Defof|Sine Integral Function}}.
{{qed}}
Category:Primitives involving Sine Function
Category:Sinc Function
Category:Sine Integral Function
nf0hi6a81xxsm3k9z75bmyrtky7esxu | :$\ds \int_{t \mathop \to 0}^{t \mathop = x} \frac {\sin t} t \rd t = \map \Si x $ | True by {{Defof|Sine Integral Function}}.
{{qed}}
[[Category:Primitives involving Sine Function]]
[[Category:Sinc Function]]
[[Category:Sine Integral Function]]
nf0hi6a81xxsm3k9z75bmyrtky7esxu | Primitive of Sinc Function | https://proofwiki.org/wiki/Primitive_of_Sinc_Function | https://proofwiki.org/wiki/Primitive_of_Sinc_Function | [
"Primitives involving Sine Function",
"Sinc Function",
"Sine Integral Function"
] | [] | [
"Category:Primitives involving Sine Function",
"Category:Sinc Function",
"Category:Sine Integral Function"
] |
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