Compact Hausdorff implies normal

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

Intermediate properties

 * Paracompact Hausdorff space:

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

Facts used

 * 1) uses::Compactness is weakly hereditary: Any closed subset of a compact space is compact.
 * 2) A union of arbitrarily many open subsets is open.
 * 3) An intersection of finitely many open subsets is open.

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

To find: Disjoint open subsets containing $$A$$ and $$x$$ respectively.

Solution:

Proof of normality
Given: Disjoint closed subsets $$A$$ and $$B$$ of $$X$$.

To find: Disjoint open subsets $$U$$ and $$V$$ of $$X$$ such that $$A \subseteq U$$ and $$B \subseteq V$$.

Solution: