Hausdorff space

A Hausdorff space (or T2 space) is a topological space $X$ such that for any distinct points $x\neq y$ there exist neighborhoods $U$ of $x$ and $V$ of $y$ with $U\cap V=\varnothing$. Equivalently, one can require $U$ and $V$ to be disjoint open sets .

Hausdorffness implies T1 and guarantees uniqueness of limits for convergent sequences ; it also ensures that compact subsets are closed .

Examples:

• Every metric space is Hausdorff.
• An infinite set with the cofinite topology is not Hausdorff.