A continuous bijection from compact to Hausdorff is a homeomorphism

Let be a Compact space, be a Hausdorff space, and be a continuous bijection. Then is a Homeomorphism. #m/thm/topology

Proof

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

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