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Hausdorff space

Definition

A topological space $(X,\mathcal{T})$is Hausdorff if the following condition (the Hausdorff condition, or 2nd separation axiom) holds: given any two distinct points $x,y \in X$, there exist disjoint open sets $U, V \subset X$ such that $x \in U$ and $y \in V$.

Properties

If $f: X \to Y$ is a continuous function, then $X$ is Hausdorff if and only if $Y$ is Hausdorff

A sequence in a Hausdorff space converges to at most one point.

If $X,Y$ are Hausdorff then so is their product space $X \times Y$ is also Hausdorff .

If $K \subset X$ is compact and $x \in X \setminus K$, then there exist disjoint open sets $U, V \subset X$ with $K \subseteq U$ and $x \in V$.

Any compact subset of a Hausdorff space is closed. This is a corollary of the point above.

For all topological spaces, a finite union of compact subsets is compact. In a Hausdorff space, a countable intersection of compact subsets is also compact.

If $X$ is compact and $Y$ is Hausdorff and $f: X \to Y$ is a continuous bijection, then $f$ is a homeomorphism.

Examples

Any metrizable space is Hausdorff.

all these pages adapted with probably insufficient credit from my university's lecture notes