avewright/proofwiki-math huggingface dataset:
Theorem: Double Pointed Finite Complement Space is not T5

:$T \times D$ is not a $T_5$ space

Proof:
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 Pointed Topologies
Category:Finite Complement Topologies
Category:Examples of T5 Spaces
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Theorem: Irrational Slope Space is not T3

Let $T$ be an irrational slope space.
Then $T$ is not a $T_3$ space.

Proof:
{{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 space.
From that contradiction it follows that our assumption that $T$ is a $T_3$ space must be false.
Hence the result by Proof by Contradiction.
{{qed}}
Theorem: Irrational Slope Space is not Regular

Let $T$ be an irrational slope space.
Then $T$ is not a regular space.

Proof:
{{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 result by Proof by Contradiction.
{{qed}}
Category:Irrational Slope Topology
Category:Examples of Regular Spaces
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Theorem: Irrational Slope Space is not Normal

Let $T$ be an irrational slope space.
Then $T$ is not a normal space.

Proof:
{{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 Contradiction.
{{qed}}
Category:Irrational Slope Topology
Category:Examples of Normal Spaces
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Theorem: Irrational Slope Space is not T4

Let $T$ be an irrational slope space.
Then $T$ is not a $T_4$ space.

Proof:
{{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 normal space.
From that contradiction it follows that our assumption that $T$ is a $T_4$ space must be false.
Hence the result by Proof by Contradiction.
{{qed}}
Theorem: Irrational Slope Space is T1

Let $T$ be an irrational slope space.
Then $T$ is a $T_1$ space.

Proof:
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
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Theorem: Irrational Slope Space is T0

Let $T$ be an irrational slope space.
Then $T$ is a $T_0$ space.

Proof:
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
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Theorem: Irrational Slope Space is not T5

Let $T$ be an irrational slope space.
Then $T$ is not a $T_5$ space.

Proof:
{{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}}
Theorem: Tychonoff Plank is not T5

Let $T$ be a Tychonoff plank.
Then $T$ is not a $T_5$ space.

Proof:
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 Completely Normal, $T$ is not a completely normal space.
Hence the result by Proof by Contradiction.
{{qed}}
Category:Tychonoff Plank
Category:Examples of T5 Spaces
0b244aeeudu7jqsxyidsfmrwwdtbnkr
Theorem: Tychonoff Plank is T1

Let $T$ be a Tychonoff plank.
Then $T$ is a $T_1$ space.

Proof:
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
Theorem: 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.

Proof:
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}}
Theorem: Sequentially Compact Space is not necessarily Compact

A sequentially compact space is not necessarily also a compact space.

Proof:
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.
{{qed}}
Theorem: Countably Compact Space is not necessarily Compact

A countably compact topological space is not necessarily also a compact space.

Proof:
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 Unit Interval is Countably Compact Space, $T$ is a countably compact space.
From Cartesian Product of Open Ordinal Space with Uncountable Cartesian Product of Closed Unit Interval is not Compact Space, $T$ is not a compact space.
Hence the result.
{{qed}}
Theorem: 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.

Proof:
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}}
Theorem: 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.

Proof:
{{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 space cannot be weakly $\sigma$-locally compact.
{{qed}}
Category:Discrete Topologies
Category:Examples of Weakly Sigma-Locally Compact Spaces
Category:Uncountable Sets
k8f0nkozjvzoyqyqu4gpqkom04wbhim
Theorem: 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.

Proof:
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}}
Theorem: 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.

Proof:
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}}
Theorem: 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.

Proof:
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
kjubvlzejbo68zzhd0f8aygfat8trh9
Theorem: 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.

Proof:
{{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 particular point space cannot be weakly $\sigma$-locally compact.
{{qed}}
Category:Particular Point Topologies
Category:Examples of Weakly Sigma-Locally Compact Spaces
Category:Uncountable Sets
infqyki6ir4eq5jeihbxt2t33qc8m0y
Theorem: Discrete Space is Weakly Locally Compact

Let $T = \struct {S, \tau}$ be a discrete topological space.
Then $T$ is weakly locally compact.

Proof:
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
Theorem: Weakly Locally Compact Space is not necessarily Sigma-Compact

A weakly locally compact topological space is not necessarily also a $\sigma$-compact space.

Proof:
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
Category:Sequence of Implications of Local Compactness Properties
78p8fsv8bw3ygg14utcntfkfqljxh2j
Theorem: Weakly Locally Compact Space is not necessarily Lindelöf

A weakly locally compact topological space is not necessarily also a Lindelöf space.

Proof:
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 Implications of Local Compactness Properties
45gei8tyqswgkr3s5reqkxhqfe2r01t
Theorem: Weakly Sigma-Locally Compact Space is not necessarily Compact

A weakly $\sigma$-locally compact topological space is not necessarily also a compact space.

Proof:
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-Locally Compact Spaces
Category:Sequence of Implications of Local Compactness Properties
bch3zdfomwt6hvf01oarphexgkwb93y
Theorem: Nested Interval Space is not Compact

Let $T = \struct {S, \tau}$ be a nested interval space.
Then $T$ is ''not'' a compact space.

Proof:
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
Theorem: Strongly Locally Compact Space is not necessarily Lindelöf

A strongly locally compact topological space is not necessarily also a Lindelöf space.

Proof:
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 of Implications of Local Compactness Properties
r6lq9mm5l6qti52b8g4husiifoth06k
Theorem: Strongly Locally Compact Space is not necessarily Compact

A strongly locally compact topological space is not necessarily also a compact space.

Proof:
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 of Implications of Local Compactness Properties
05fr6w34kepwspwxdw85ugbj6cu0it5
Theorem: Sigma-Compact Space is not necessarily Weakly Sigma-Locally Compact

A $\sigma$-compact space is not necessarily also a weakly $\sigma$-locally compact space.

Proof:
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:Weakly Sigma-Locally Compact Spaces
Category:Sequence of Implications of Local Compactness Properties
atmyqu6764q2pn7hkimotdf3007auuc
Theorem: Evenly Spaced Integer Space is Sigma-Compact

The evenly spaced integer space is a $\sigma$-compact space.

Proof:
We have Evenly Spaced Integer Space is Countably Infinite.
The result follows from Countable Space is $\sigma$-Compact.
{{qed}}
Theorem: Evenly Spaced Integer Space is Countably Infinite

The evenly spaced integer space is a countably infinite topological space.

Proof:
By its definition, $S$ is the set of integers $\Z$.
The result follows from Integers are Countably Infinite.
{{qed}}
Theorem: Evenly Spaced Integer Space is not Weakly Locally Compact

The evenly spaced integer space is not a weakly locally compact space.

Proof:
{{Recall|Weakly Locally Compact Space|weakly locally compact space}}
{{:Definition:Weakly Locally Compact Space}}
{{ProofWanted}}
Theorem: Evenly Spaced Integer Space is not Weakly Sigma-Locally Compact

The evenly spaced integer space is not a weakly $\sigma$-locally compact space.

Proof:
{{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}}
Category:Evenly Spaced Integer Topology
Category:Examples of Weakly Sigma-Locally Compact Spaces
tbmf1h06g75tnhbk7hss4w81j57lsxl
Theorem: Dieudonné Plank is not Paracompact

Let $T = \struct {S, \tau}$ be the Dieudonné plank.
Then $T$ is not paracompact.

Proof:
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
Theorem: Metacompact Space is not necessarily Countably Paracompact

A metacompact topological space is not necessarily also a paracompact space.

Proof:
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 Paracompactness Properties
rxefkje1bvzokyelzu2m629592mwwn5
Theorem: Countably Paracompact Space is not necessarily Paracompact

A countably paracompact topological space is not necessarily also a paracompact space.

Proof:
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}}
Theorem: Countably Metacompact Space is not necessarily Metacompact

A countably metacompact topological space is not necessarily also a metacompact space.

Proof:
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 Paracompactness Properties
kbrzduowb7ifn4vp2cyuqv18rjz83oc
Theorem: Countably Compact Space is not necessarily Metacompact

A countably compact topological space is not necessarily also a metacompact space.

Proof:
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 the result.
{{qed}}
Category:Countably Compact Spaces
Category:Metacompact Spaces
Category:Sequence of Implications of Paracompactness Properties
ji8u0ishsa7ewp2ygydky3y1eczurr1
Theorem: 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 $\CC$.

Proof:
{{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 neither $\set 1$ nor $\set 2$ are elements of $\set {\set {1, 2} }$.
Hence it is not the case that $\RR \subseteq \CC$.
Hence the result.
{{qed}}
Theorem: Global Compactness Properties which are Weakly Hereditary

The following global compactness properties are weakly hereditary:
:Compactness
:Countable Compactness
{{WIP}}

Proof:
We have:
* Closed Subspace of Compact Space is Compact
* Countable Compactness is Weakly Hereditary
{{ProofWanted}}
Theorem: Separability is Preserved only in Open Subspace

The property of separability is preserved only in subspaces which are open.

Proof:
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 $K \notin \tau$.
We will show that $K$ is not necessarily a separable space.
{{ProofWanted}}
Theorem: 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.

Proof:
{{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 $T_5$ space.
We have shown that $T$ is both a $T_5$ space and a $T_1$ space.
Hence the result.
{{qed}}
Theorem: 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.

Proof:
We are given that $T = \struct {S, \tau}$ is a $T_2$ (Hausdorff) space.
Theorem: 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.

Proof:
{{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}}
Theorem: 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.

Proof:
{{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}}
Theorem: 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.

Proof:
{{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}}
Theorem: 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.

Proof:
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 Compactness Properties in Hausdorff Space
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Theorem: 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.

Proof:
{{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.
{{qed}}
Theorem: 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.

Proof:
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}}
Theorem: 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.

Proof:
{{ProofWanted|There will probably be a counterexample that can be found if we search hard enough}}
Theorem: Sorgenfrey Line is Paracompact

The Sorgenfrey line is a paracompact space.

Proof:
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}}