Characterization of T0 Space by Closed Sets

Theorem

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

Then

for every points $x, y \in S$ if $x \ne y$ then

there exists a closed subset $F$ of $S$ such that $x \in F$ and $y \notin F$

or

there exists a closed subset $F$ of $S$ such that $x \notin F$ and $y \in F$

Let $T$ be a $T_0$ space.

Let $x, y \in S$ such that

$x \ne y$

By definition of $T_0$ space:

$\left({\exists U \in \tau: x \in U \land y \notin U}\right) \lor \left({\exists U \in \tau: x \notin U \land y \in U}\right)$

WLOG: Suppose:

$\exists U \in \tau: x \in U \land y \notin U$

By definition:

$\complement_S\left({U}\right)$ is closed

where $\complement_S\left({U}\right)$ denotes the relative complement of $U$ in $S$.

$x \notin \complement_S\left({U}\right) \land y \in \complement_S\left({U}\right)$

Thus:

$\exists F \subseteq S: F$ is closed $\land\, x \notin F \land y \in F$

$\Box$

This statement follows mutatis mutandis.

$\blacksquare$