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

Question

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?

Answer 1

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.

Answer 2

Hint: That map is open that means that image of an open set is open.