avewright/proofwiki-math huggingface dataset:
Theorem: Completely Regular Property is Not Preserved under Expansion

If $T_A$ is a completely regular space, then it is not necessarily the case that so is $T_B$.

Proof:
Let $T_A$ be a completely regular space.
From Completely Regular Space is Regular, it follows that $T_A$ is a regular space.
But from Regular Property is Not Preserved under Expansion, it is not necessarily the case that $T_B$ is also a regular space.
But again, we have:
:Completely Regular Space is Regular
Hence, by the Rule of Transposition, it is not necessarily the case that $T_B$ is a completely regular space.
Hence the result.
{{qed}}
Category:Completely Regular Spaces
Category:Separation Properties Not Preserved under Expansion
mq3v5qrgpzk0werw2tsttxsejke852o
Theorem: Normal Property is Not Preserved under Expansion

If $T_A$ is a normal space, then it is not necessarily the case that so is $T_B$.

Proof:
Let $T_A$ be a normal space.
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
Thus $T_A$ is a $T_4$ space.
From $T_1$ Property is Preserved under Expansion we have that $T_B$ is a $T_1$ space.
From $T_4$ Property is Not Preserved under Expansion, it is not necessarily the case that $T_B$ is also a $T_4$ space.
Hence it is not necessarily the case that $T_B$ is also a normal space.
Hence the result.
{{qed}}
Category:Normal Spaces
Category:Separation Properties Not Preserved under Expansion
n7qa4skckvcd34b4zivdjctftg51ftt
Theorem: Completely Normal Property is Not Preserved under Expansion

If $T_A$ is a completely normal space, then it is not necessarily the case that so is $T_B$.

Proof:
Let $T_A$ be a completely normal space.
{{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
Thus $T_A$ is a $T_5$ space.
From $T_1$ Property is Preserved under Expansion we have that $T_B$ is a $T_1$ space.
But from $T_5$ Property is Not Preserved under Expansion, it is not necessarily the case that $T_B$ is also a $T_5$ space.
Hence it is not necessarily the case that $T_B$ is also a completely normal space.
Hence the result.
{{qed}}
Category:Completely Normal Spaces
Category:Separation Properties Not Preserved under Expansion
j8xqqr7mzs8f85pgcar0ngh8b3tmkwc
Theorem: Indiscrete Extension of Reals is not Regular

Let $T$ be an indiscrete extension of $\R$.
Then $T$ is not a regular space.

Proof:
{{Recall|Regular Space|index = 1}}
{{:Definition:Regular Space/Definition 1}}
From Indiscrete Extension of Reals is $T_0$, we have that $T$ is a $T_0$ space.
From Indiscrete Extension of Reals is not $T_3$, we have that $T$ is not a $T_3$ space.
Hence the result.
{{qed}}
Category:Indiscrete Extensions of Reals
Category:Examples of Regular Spaces
hjyvzimxdyoamvtayguyfku0pgj0ct3
Theorem: Indiscrete Extension of Reals is not Normal

Let $T$ be an indiscrete extension of $\R$.
Then $T$ is not a normal space.

Proof:
{{Recall|Normal Space|normal space|index = 1}}
{{:Definition:Normal Space/Definition 1}}
From Indiscrete Extension of Reals is $T_1$, we have that $T$ is a $T_1$ space.
From Indiscrete Extension of Reals is not $T_4$, we have that $T$ is not a $T_4$ space.
Hence the result.
{{qed}}
Category:Indiscrete Extensions of Reals
Category:Examples of Normal Spaces
707u4jxodj0wtbb0rfyinknioljdoyb
Theorem: Indiscrete Extension of Reals is not Completely Normal

Let $T$ be an indiscrete extension of $\R$.
Then $T$ is not a completely normal space.

Proof:
{{Recall|Completely Normal Space|completely normal space|index = 1}}
{{:Definition:Completely Normal Space/Definition 1}}
From Indiscrete Extension of Reals is $T_1$, we have that $T$ is a $T_1$ space.
From Indiscrete Extension of Reals is not $T_5$, we have that $T$ is not a $T_5$ space.
Hence the result.
{{qed}}
Category:Indiscrete Extensions of Reals
Category:Examples of Completely Normal Spaces
su8rsutwrah7h7lgjndzp1dfa0h7gz4
Theorem: Real Number Line with Euclidean Topology is T0

:$\struct {\R, \tau_d}$ is a $T_0$ space

Proof:
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 $T_0$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T0 Spaces
4b7xcpig775xg62ukcxkxbg602bsgj2
Theorem: Real Number Line with Euclidean Topology is T1

:$\struct {\R, \tau_d}$ is a $T_1$ space

Proof:
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 $T_1$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T1 Spaces
s5psh1ltgr499skda9ljeuxurwyuklb
Theorem: Real Number Line with Euclidean Topology is T2

:$\struct {\R, \tau_d}$ is a $T_2$ (Hausdorff) space

Proof:
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 $T_2$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Hausdorff Spaces
i2s3u94unr54d7qxohjnp2sqjtpbaix
Theorem: Real Number Line with Euclidean Topology is Semiregular

:$\struct {\R, \tau_d}$ is a semiregular space

Proof:
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 Semiregular.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Semiregular Spaces
9e3zco4a73jmwfwzeogs55md84tb3lw
Theorem: Real Number Line with Euclidean Topology is T2.5

:$\struct {\R, \tau_d}$ is a $T_{2 \frac 1 2}$ space

Proof:
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 $T_{2 \frac 1 2}$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T2.5 Spaces
n6fkfznajemthe1oj16b99rj9p6xga1
Theorem: Real Number Line with Euclidean Topology is T3.5

:$\struct {\R, \tau_d}$ is a $T_{3 \frac 1 2}$ space

Proof:
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 $T_{3 \frac 1 2}$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T3.5 Spaces
1p626zebtq7mg1gaqhenejcw3fn2s9v
Theorem: Real Number Line with Euclidean Topology is T3

:$\struct {\R, \tau_d}$ is a $T_3$ space

Proof:
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 $T_3$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T3 Spaces
b94laj2v76sw03q9co1aubs0w1i0w5f
Theorem: Real Number Line with Euclidean Topology is T4

:$\struct {\R, \tau_d}$ is a $T_4$ space.

Proof:
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 $T_4$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T4 Spaces
lm69reqvk0umit8cyopendwvhlvh0rn
Theorem: Real Number Line with Euclidean Topology is T5

:$\struct {\R, \tau_d}$ is a $T_5$ space

Proof:
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 $T_5$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of T5 Spaces
k7vm1tcvziwgzy6jxn936xlvpjh15vi
Theorem: Real Number Line with Euclidean Topology is Perfectly T4

:$\struct {\R, \tau_d}$ is a perfectly $T_4$ space

Proof:
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 $T_4$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Perfectly T4 Spaces
8tum0dfuah4x8cqih0mbjpmiks9fxa2
Theorem: Real Number Line with Euclidean Topology is Fully T4

:$\struct {\R, \tau_d}$ is a fully $T_4$ space.

Proof:
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 $T_4$.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Fully T4 Spaces
qsuy1ssbkz5a3k510d9o3fbmlblx9ui
Theorem: Real Number Line with Euclidean Topology is Regular

:$\struct {\R, \tau_d}$ is a regular space

Proof:
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 Regular.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Regular Spaces
kdkvnn7hqh3wy5ui8ahaql2brl21z0n
Theorem: Real Number Line with Euclidean Topology is Urysohn

:$\struct {\R, \tau_d}$ is an Urysohn space

Proof:
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 Urysohn.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Urysohn Spaces
pgug638hzoo0rsqji288zd6kges68mw
Theorem: Real Number Line with Euclidean Topology is Completely Regular

:$\struct {\R, \tau_d}$ is a completely regular space

Proof:
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 Completely Regular.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Completely Regular Spaces
cdhffaunv2o17lipke1ul26lg0e0swe
Theorem: Real Number Line with Euclidean Topology is Completely Normal

:$\struct {\R, \tau_d}$ is a completely normal space

Proof:
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 Completely Normal.
{{qed}}
Category:Real Number Line satisfies all Separation Axioms
Category:Real Number Line with Euclidean Topology
Category:Examples of Completely Normal Spaces
llnsc1nrewqkk9nxst3vwwh03msvdss
Theorem: Real Number Line with Euclidean Topology is Fully Normal

:$\struct {\R, \tau_d}$ is a fully normal space

Proof:
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
dvh8w3vacwgdfk35saewcgof9r8uwhw
Theorem: Real Number Line with Euclidean Topology is Perfectly Normal

:$\struct {\R, \tau_d}$ is a perfectly normal space

Proof:
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 Spaces
99yncbie87rxrxvsopciea2xs04yy40
Theorem: Real Number Line with Euclidean Topology is Normal

:$\struct {\R, \tau_d}$ is a normal space

Proof:
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
seofn6xolwmzqq18rcvfny9pxds38vc
Theorem: Indiscrete Extension of Reals is T1

Let $T$ be an indiscrete extension of $\R$.
Then $T$ is a $T_1$ space.

Proof:
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
Category:Examples of T1 Spaces
999ja1mkp4h9x80nwq1pjoiai0m7iyk
Theorem: Indiscrete Extension of Reals is T0

Let $T$ be an indiscrete extension of $\R$.
Then $T$ is a $T_0$ space.

Proof:
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
Category:Examples of T0 Spaces
bv83zmr1qivrhyig2gbjlartuzq6k53
Theorem: Regular Property is Hereditary

:The property of being a regular space is hereditary.

Proof:
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}}
Theorem: Completely Regular Property is Hereditary

:The property of being a completely regular space is hereditary.

Proof:
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}}
Theorem: Completely Normal Property is Hereditary

:The property of being a completely normal space is hereditary.

Proof:
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.
{{qed}}
Theorem: 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.

Proof:
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 a subspace.
Hence the result.
{{qed}}
Theorem: T2.5 Space is not necessarily Urysohn

A $T_{2 \frac 1 2}$ topological space is not necessarily an Urysohn space.

Proof:
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}}
Theorem: Double Pointed Finite Complement Space is not T0

:$T \times D$ is not a $T_0$ space

Proof:
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
Theorem: 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.

Proof:
{{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}}
Theorem: 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.

Proof:
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 Indiscrete Non-Singleton Space is not $T_1$, $D$ is not a $T_1$ space.
The result follows by Proof by Contradiction.
{{qed}}
Theorem: 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.

Proof:
{{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
gw5tazfr65gvdblkcanm3laf045fkwv
Theorem: 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.

Proof:
{{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
7ixbsfulkhgky4i2b7s8s2peuqiebfn
Theorem: 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.

Proof:
{{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}}
Theorem: 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.

Proof:
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 Indiscrete Non-Singleton Space is not $T_2$, $D$ is not a $T_2$ space.
The result follows by Proof by Contradiction.
{{qed}}
Theorem: Double Pointed Finite Complement Space is not T1

:$T \times D$ is not a $T_1$ space

Proof:
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
Theorem: Double Pointed Finite Complement Space is not T2

:$T \times D$ is not a $T_2$ space

Proof:
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
rmpzrhkxwzsx0hc9ymkdk65hty9lvki
Theorem: Double Pointed Finite Complement Space is not T3

:$T \times D$ is not a $T_3$ space

Proof:
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 Pointed Topologies
Category:Finite Complement Topologies
Category:Examples of T3 Spaces
s0rexmbrlbag16c83eoufwgz93jkx8b
Theorem: 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.

Proof:
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
Theorem: 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.

Proof:
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 $T_2$ space.
From this contradiction it follows that $T$ is not a $T_3$ space.
{{qed}}
Theorem: 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.

Proof:
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 a $T_4$ space.
{{qed}}
Theorem: 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.

Proof:
{{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}}
Theorem: 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.

Proof:
{{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 Pointed Topologies
Category:Examples of T2.5 Spaces
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Theorem: Double Pointed Finite Complement Space is not T4

:$T \times D$ is not a $T_4$ space

Proof:
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 Pointed Topologies
Category:Finite Complement Topologies
Category:Examples of T4 Spaces
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