Hausdorff space - Imported Knowlpedia Content

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 $U\cap V=\varnothing$ .

Equivalent characterizations

Equivalently, one can require $U$ and $V$ to be disjoint open sets.

Remarks

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.