Problem of the Week #23 - November 5th, 2012

[Chris L T521]

Here's this week's problem.

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Problem: Let $f:X\rightarrow Y$ be a bijective continuous function between topological spaces $X$ and $Y$. If $X$ is compact and $Y$ is Hausdorff, show that $f$ is a homeomorphism.

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[Chris L T521]

No one answered this week's question. Here's my solution.

Spoiler

Proof: All we really need to do is justify that $f^{-1}:Y\rightarrow X$ is continuous; to do this, we need to show that the images of closed sets in $X$ are closed in $Y$. Let $A\subseteq X$ be a closed set in $X$. Since $X$ is compact, it follows that $A$ is compact (since a set is compact iff it's closed and bounded). The image of a compact set is compact, so it follows that $f(A)$ is compact. Since compact sets in a Hausdorff space are closed, it now follows that $f(A)$ is closed in $Y$ and thus $f^{-1}$ is continuous. Q.E.D.