nLab closed subspaces of compact Hausdorff spaces are equivalently compact subspaces

Statement

Proposition

Let $(X,\tau)$ be a compact Hausdorff topological space and let $Y \subset X$ be a topological subspace. Then the following are equivalent:

1. $Y \subset X$ is a closed subspace;

2. $Y$ is a compact topological space.

Proof

The two directions to be proven are

1. closed subsets of compact spaces are compact

2. compact subspaces of Hausdorff spaces are closed.

See the proofs there.

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