T2 (Hausdorff)

Every compact subspace of a Hausdorff space is closed. That is, if \(Y\) is a compact subspace of a \(T_2\) space \(X\), then \(Y\) is a closed subset of \(X\).

(VUT) A continuous bijection from a compact space to a Hausdorff space is a homeomorphism.

Compact Hausdorff spaces are \(T_4\).

\(T_2\) is not preserved by quotients, continuous images, or continuous preimages.

Let \(X\) be a \(T_2\) topological space and let \(p \in X\). Then \(\{p\}\) is closed in \(X\).