[[Compact space]]
# 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 

> [!check]- Proof
> Since [[the continuous image of a compact space is compact]], $Y$ is compact.
> If $A \in X$ is closed, [[Closed subsets of a compact space are compact|then it is also compact]], and thus its image $fA$ [[The continuous image of a compact space is compact|is also compact]], [[Closed subsets of a compact space are compact|whence it is closed]].
> Thence $f$ is a [[Open and closed maps|closed map]] and [[A bijection is open iff it is closed|therefore an open map]].
> Therefore $f$ is an open continuous bijection, i.e. a [[Homeomorphism]].
> <span class="QED"/>

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