A continuous bijection from a compact space to a $T_2$ space is always a homeomorphism

general-topology compactness continuity separation-axioms

Solution 1:

Hint: You only have to prove that it's a closed map.

Now, if you take a closed subspace $A\subseteq X$ you know it is (BLANK) and so its image is (BLANK). But, by Hausdorffness, (BLANK) subspaces are closed.

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