Compact Hausdorff Space is T4

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Theorem

Let $T = \left({S, \tau}\right)$ be a compact Hausdorff space.


Then $T$ is a $T_4$ space.


Proof

We have that a Compact Subspace of Hausdorff Space is Closed.

We also have that a Closed Subspace of Compact Space is Compact.

We also have that Disjoint Compact Sets in Hausdorff Space have Disjoint Neighborhoods.

$T$ is a $T_4$ space when any two disjoint closed subsets of $S$ are separated by neighborhoods.

Hence the result.

$\blacksquare$


Sources