Definition:Hausdorff Space

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

Definition 1
$\left({S, \tau}\right)$ is a Hausdorff space or $T_2$ space iff:
 * $\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \varnothing$

That is, for any two distinct points $x, y \in S$ 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 iff every two points in $S$ are separated by open sets.

This condition is known as the Hausdorff condition.

For short, we can say $T$ is Hausdorff, using the name as an adjective.

Conveniently, a topological space is Hausdorff if any two distinct points can be housed off from one another in separate disjoint open sets.

Definition 2
$\left({S, \tau}\right)$ is a Hausdorff space or $T_2$ space iff each point is the intersection of all its closed neighborhoods.

Equivalence of Definitions
See Equivalent Definitions for $T_2$ Space for a proof that the two definitions are equivalent.

Also see

 * Basic Properties of a Hausdorff Space