# 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 $T_{1}$-space if and only if sets $\{x\}$, $\{y\}$ are separated for all distinct $x,y\in X$.

###### Proof.

Suppose $X$ is a $T_{1}$-space. Then every singleton is closed and if $x,y\in X$ are distinct, then

 $\displaystyle\{x\}\cap\overline{\{y\}}$ $\displaystyle=$ $\displaystyle\{x\}\cap\{y\}=\emptyset,$ $\displaystyle\overline{\{x\}}\cap\{y\}$ $\displaystyle=$ $\displaystyle\{x\}\cap\{y\}=\emptyset,$

and $\{x\}$, $\{y\}$ are separated. On the other hand, suppose that $\{x\}\cap\overline{\{y\}}=\emptyset$ for all $x\neq y$. It follows that $\overline{\{y\}}=\{y\}$, so $\{y\}$ is closed and $X$ is a $T_{1}$-space. ∎

Title a space is $T_{1}$ if and only if distinct points are separated ASpaceIsT1IfAndOnlyIfDistinctPointsAreSeparated 2013-03-22 15:16:49 2013-03-22 15:16:49 matte (1858) matte (1858) 6 matte (1858) Theorem msc 54D10