A Hausdorff space separates any two distinct points by disjoint open neighborhoods.
Every compact subset of a Hausdorff space is closed. For a point outside the compact set, separate it from each point of the set and use a finite subcover to obtain one neighbourhood disjoint from the whole set.
A continuous bijection from a compact space to a Hausdorff space is a homeomorphism. It maps closed sets to compact sets, which are closed in the codomain, so it is a closed map and its inverse is continuous.
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A Hausdorff space, also known as a \(T_2\) space, is a type of topological space that satisfies a particular separation property.