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A continuous map from a compact space to a Hausdorff space is closed. Why this is true?

Help me please I want to learn why this is correct.

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up vote 9 down vote accepted

Combine the following facts:

1) A closed subspace of a compact space is compact.

2) A continuous map always maps compact spaces onto compact spaces.

3) Compact subspaces of Hausdorff spaces are closed.

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Thanks @Martin Brandenburg :)) – Aera Apr 15 '13 at 18:23
Is there any reason to talk about subspaces instead of subsets here? – Marc van Leeuwen Apr 15 '13 at 18:46
Yes, but the comment box is too small for a digression on category theory ;). Sets have no topology, therefore "compact set" is absolute nonsense (but unfortunately quite common, sigh). – Martin Brandenburg Apr 16 '13 at 10:05

If $f:X\to Y$ is continuous and $K\subseteq X$ is compact then $f(K)$ is compact.

Now use that closed subsets of compact sets are compacts and that compact subsets of Hausdorff spaces are closed.

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