Hausdorff space

A topological space $(X,\tau)$ is said to be $T_2$ (or said to satisfy the $T_2$ axiom) if given distinct $x,y\in X$ , there exist disjoint open sets $U,V\in\tau$ (that is, $U\cap V=\emptyset$ ) such that $x\in U$ and $y\in V$ .

A $T_2$ space is also known as a Hausdorff space. A Hausdorff topology for a set $X$ is a topology $\tau$ such that $(X,\tau)$ is a Hausdorff space.

Properties

1. $X$ is a Hausdorff space.
2. The set $$ \Delta=\{(x,y)\in X\times X:x=y\} $$ is closed in the product topology of $X\times X$ .
3. For all $x\in X$ , we have $$ \{x\} = \bigcap \{A : A\subseteq X\ \mbox{closed}, \mbox{$\exists$ open set}\ U\ \mbox{such that}\ x\in U\subseteq A\}. $$

Important examples of Hausdorff spaces are metric spaces, manifolds, and topological vector spaces.