# Definition:Hausdorff Space/Definition 1

## Definition

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

$\left({S, \tau}\right)$ 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.

That is:

$\left({S, \tau}\right)$ is a $T_2$ space if and only if every two elements in $S$ are separated by open sets.

## Source of Name

This entry was named for Felix Hausdorff.