Definition:Hausdorff Space/Definition 3

Definition
Let $T = \struct {S, \tau}$ be a topological space.

$\struct {S, \tau}$ is a Hausdorff space or $T_2$ space :
 * $\forall x, y \in S, x \ne y: \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x \cap N_y = \varnothing$

That is:
 * for any two distinct elements $x, y \in S$ there exist disjoint neighborhoods $N_x, N_y \subseteq S$ containing $x$ and $y$ respectively.

That is:
 * $\struct {S, \tau}$ is a $T_2$ space every two elements in $S$ are separated by neighborhoods.

Also see

 * Equivalence of Definitions of $T_2$ Space