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

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?

• – user228113 May 24 '16 at 22:16

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.
• 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. – user228113 May 24 '16 at 22:22