\begin{definition}[Definition:Hausdorff Space/Definition 1]
Let $T = \struct {S, \tau}$ be a topological space.
$\struct {S, \tau}$ is a '''Hausdorff space''' or '''$T_2$ space''' {{iff}}:
:$\forall x, y \in S, x \ne y: \exists U, V \in \tau: x \in U, y \in V: U \cap V = \O$ 
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:
:$\struct {S, \tau}$ is a '''$T_2$ space''' {{iff}} every two elements in $S$ are separated by open sets.
\end{definition}