Equivalence of Definitions of T2 Space

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Theorem

The following definitions of the concept of $T_2$ (Hausdorff) space are equivalent:


Let $T = \left({S, \tau}\right)$ be a topological space.

Definition 1

$\left({S, \tau}\right)$ is a Hausdorff space or $T_2$ space if and only if:

$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \varnothing$

That is:

for any two distinct elements $x, y \in S$ there exist disjoint open sets $U, V \in \tau$ containing $x$ and $y$ respectively.

Definition 2

$\left({S, \tau}\right)$ is a Hausdorff space or $T_2$ space if and only if each point is the intersection of all its closed neighborhoods.

Definition 3

$\left({S, \tau}\right)$ is a Hausdorff space or $T_2$ space if and only if:

$\forall x, y \in S, x \ne y: \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x \cap N_y = \varnothing$

That is:

for any two distinct elements $x, y \in S$ there exist disjoint neighborhoods $N_x, N_y \subseteq S$ containing $x$ and $y$ respectively.


Proof

Definition 1 implies Definition 2

Let $T = \left({S, \tau}\right)$ be a topological space for which:

$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \varnothing$


Let us take any arbitrary $x, y \in S: x \ne y$.

Let $\mathcal C_x$ be the set of all closed neighborhoods of $x$:

$\mathcal C_x = \left\{{H: \complement_S \left({H}\right) \in \tau, \exists U \in \tau: x \in U \subseteq H}\right\}$

where $\complement_S \left({H}\right)$ is the complement of $H$ in $S$.


We need to demonstrate that the only element in the intersection of $\mathcal C_x$ is $x$:

$\bigcap \mathcal C_x = \left\{{x}\right\}$

and to do that we show that if $y \ne x$ then $y \notin \bigcap \mathcal C_x$.


Let $C = \bigcap C_x$.

Clearly $x \in C$ and so $\left\{{x}\right\} \subseteq C_x$.

We have that $\exists U, V \in \tau: x \in U, y \in V: U \cap V = \varnothing$ by hypothesis.

As $x \in U$ it follows that $x \notin V$ and so $x \in \complement_S \left({V}\right)$.

Thus $x \in U \subseteq \complement_S \left({V}\right)$.

That is, $\complement_S \left({V}\right)$ is a closed neighborhood of $x$ and so $\complement_S \left({V}\right) \in \mathcal C_x$.


As $y \in V$ it follows that $y \notin \complement_S \left({V}\right)$.

So $\complement_S \left({V}\right)$ is a closed neighborhood of $x$ which does not contain $y$.

So $y \notin \bigcap C_x$.


As $y$ is arbitrary:

$\forall y \in S, y \ne x: \exists H: \complement_S \left({H}\right) \in \tau: y \notin H$

and so $C_x \subseteq \left\{{x}\right\}$.

That is:

$\displaystyle \forall x \in S: \left\{{x}\right\} = \bigcap \left\{{H: \complement_S \left({H}\right) \in \tau, \exists U \in \tau: x \in U \subseteq H}\right\}$

or, each point is the intersection of all its closed neighborhoods.

$\Box$


Definition 2 implies Definition 1

Let $T = \left({S, \tau}\right)$ be a topological space for which each point is the intersection of all its closed neighborhoods.


Let $x, y \in S: x \ne y$.

Let $\mathcal C_x$ be the set of all closed neighborhoods of $x$:

$\mathcal C_x = \left\{{H: \complement_S \left({H}\right) \in \tau, \exists U \in \tau: x \in U \subseteq H}\right\}$

where $\complement_S \left({H}\right)$ is the complement of $H$ in $S$.

This arises from the definition of a closed set as the complement in $S$ of an open set.


We have that:

$\displaystyle \left\{{x}\right\} = \bigcap \left\{{H: \complement_S \left({H}\right) \in \tau, \exists U \in \tau: x \in U \subseteq H}\right\}$

Then as $y \notin \left\{{x}\right\}$ it is not the case that $\forall H \in C_x: y \in H$.

So for some $H \in C_x$ it must be the case that $y \in \complement_S \left({H}\right) = V$.


But $V = \complement_S \left({H}\right) \in \tau$, that is, $V$ is open in $T$.

Also, as $U \subseteq H$, it must follow that $U \cap V = \varnothing$.

So:

$\exists U, V \in \tau: x \in U, y \in V: U \cap V = \varnothing$


As $x$ and $y$ are arbitrary, it follows that:

$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \varnothing$

$\Box$


Definition 1 implies Definition 3

Let $T = \left({S, \tau}\right)$ be a topological space for which:

$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \varnothing$


Let $x, y \in S: x \ne y$ be arbitrary.

From Set is Open iff Neighborhood of all its Points, $U$ and $V$ are neighborhoods of $x$ and $y$.

Thus as $x$ and $y$ are arbitrary:

$\forall x, y \in S, x \ne y: \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x \cap N_y = \varnothing$

$\Box$


Definition 3 implies Definition 1

Let $T = \left({S, \tau}\right)$ be a topological space for which:

$\forall x, y \in S, x \ne y: \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x \cap N_y = \varnothing$


Let $x, y \in S: x \ne y$ be arbitrary.

We have that:

$\exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x \cap N_y = \varnothing$


Aiming for a contradiction, suppose $\exists z \in S: z \in U \cap V$.

Then $z \in U, z \in V$

\(\displaystyle z\) \(\in\) \(\displaystyle U\) Definition of Set Intersection
\(\, \displaystyle \land \, \) \(\displaystyle z\) \(\in\) \(\displaystyle V\) Definition of Set Intersection
\(\displaystyle \implies \ \ \) \(\displaystyle z\) \(\in\) \(\displaystyle N_x\) Definition of Subset
\(\, \displaystyle \land \, \) \(\displaystyle z\) \(\in\) \(\displaystyle N_y\) Definition of Subset
\(\displaystyle \implies \ \ \) \(\displaystyle z\) \(\in\) \(\displaystyle N_x \cap N_y\) Definition of Set Intersection

From this contradiction it follows that $U \cap V = \varnothing$.

As $x$ and $y$ are arbitrary, it follows that:

$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \varnothing$

$\blacksquare$


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