a space is $T_1$ if and only if distinct points are separated

Theorem 1.

Let $X$ be a topological space. Then $X$ is a $T_{\mathrm{1}}$-spaceif and only if sets $\mathrm{\{}x\mathrm{\}}$, $\mathrm{\{}y\mathrm{\}}$are separated for all distinct $x\mathrm{,}y\mathrm{\in }X$.

Proof.

Suppose $X$ is a $T_1$-space. Then every singleton isclosed and if $x,y\in X$ are distinct, then

$\mathrm{\{}x\}\cap \overline{\{y\}}$$=$$\mathrm{\{}x\}\cap \{y\}=\mathrm{\varnothing },$

$\overline{\{x\}}\cap \{y\}$$=$$\mathrm{\{}x\}\cap \{y\}=\mathrm{\varnothing },$

and $\{x\}$, $\{y\}$ are separated.On the other hand, suppose that$\{x\}\cap \overline{\{y\}}=\mathrm{\varnothing }$for all $x\ne y$. It follows that $\overline{\{y\}}=\{y\}$, so $\{y\}$is closed and $X$ is a $T_1$-space.∎