Compact Hausdorff implies normal

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property must also satisfy the second topological space property
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Statement

Property-theoretic statement

The property of topological spaces of being compact Hausdorff implies, or is stronger than, the property of being normal.

Verbal statement

Any compact Hausdorff space (a topological space that is both a compact and Hausdorff) is normal.

Related facts

Intermediate properties

• Paracompact Hausdorff space: Further information: paracompact Hausdorff implies normal

Other related facts

• any closed subset of a compact space is compact
• any compact subset of a Hausdorff space is closed: The proof of this uses a very similar argument.
• Any locally compact Hausdorff space is completely regular

Proof

Suppose $X$ is a compact Hausdorff space. We need to show that $X$ is normal. We will proceed in two steps: we will first show that $X$ is a regular space, and then show that $X$ is normal.

Proof of regularity

Suppose $x\in X$ is a point and $A$ is a closed subset of $X$ not containing $x$.