\begin{definition}[Definition:Hausdorff Space/Definition 3]
Let $T = \struct {S, \tau}$ be a topological space.
$\struct {S, \tau}$ is a '''Hausdorff space''' or '''$T_2$ space''' {{iff}}:
:$\forall x, y \in S, x \ne y: \exists N_x, N_y \subseteq S: \exists U, V \in \tau: x \subseteq U \subseteq N_x, y \subseteq V \subseteq N_y: N_x \cap N_y = \O$
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''' {{iff}} every two elements in $S$ are separated by neighborhoods.
\end{definition}