A continuous bijection from compact to Hausdorff is a homeomorphism

Let $𝑋$ be a Compact space, $𝑌$ be a Hausdorff space, and $𝑓 : 𝑋 \to 𝑌$ be a continuous bijection. Then $𝑓$ is a Homeomorphism. #m/thm/topology

Proof

Since the continuous image of a compact space is compact, $𝑌$ is compact. If $𝐴 \in 𝑋$ is closed, then it is also compact, and thus its image $𝑓 𝐴$ is also compact, whence it is closed. Thence $𝑓$ is a closed map and therefore an open map. Therefore $𝑓$ is an open continuous bijection, i.e. a Homeomorphism.

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