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Prove that any continuous bijection $f:X \rightarrow Y$ from a compact space $X$ to a Hausdorff space $Y$ is a homeomorphism

Requirements for a homeomorphism $f:X \rightarrow Y$:

  1. $f$ is continuous
  2. $f$ is bijective
  3. $f^{-1}$ is continuous

The first two properties are given in the question, so we just need to show that the inverse is continuous.

So $f^{-1}(Y)$ is the preimage of a Hausdorff space to a compact space. Why is this continuous?

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Since the $f$ is continuous and bijective, it is a homeomorphism if and only if it is closed. But closed subset of a compact space are compact, image of compact is compact, and compact subsets of a Hausdorff space are closed.

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  • $\begingroup$ thank you. I was wondering why does it have to be closed? $\endgroup$ – thinker May 24 '16 at 22:17
  • $\begingroup$ Homeomorphism are closed. Conversely, a continuous, closed and bijective function is a homeomorphism because, if $A\subseteq X$ is closed, then by bijectivity $[f^{-1}]^{-1}(A)$ (a.k.a. the pre-image of set $A$ under the function $f^{-1}$) is $f(A)$ (a.k.a. the image of $A$ under the function $f$). That's a set-theoretical fact. But $f(A)$ is closed by hypothesis. $\endgroup$ – user228113 May 24 '16 at 22:22
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Hint: That map is open that means that image of an open set is open.

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