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We have that $f\colon X\to Y$ is a continuous function, $X$ is a compact space and $Y$ is a Hausdorff space. Prove that $f$ is a closed function.

Martin Sleziak
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Sokkun Yoem
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    It is because a closed subset of a compact space is compact, and the image of a compact space under a continuous function is compact, and finally a compact subset of a hausdorff space is closed. You need those three facts to prove this. – Gregory Grant Jun 16 '15 at 15:57

1 Answers1

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Let $C$ be a closed subset of $X$; you want to prove that $f(C)$ is closed in $Y$.

We use three basic facts about compact spaces:

  1. a closed subset of a compact space is compact;
  2. the image of a compact subset under a continuous function is compact;
  3. a compact subset of a Hausdorff space is closed.

Now the proof of your statement.

Since $X$ is compact, $C$ is compact as well; therefore $f(C)$ is compact. A compact subset of a Hausdorff space is closed. Hence $f(C)$ is closed.

egreg
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