Compact space and Hausdorff space
1,088
Solution 1
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.
Solution 2
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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Author by
Aera
Updated on August 01, 2022Comments
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Aera 10 months
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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Aera about 10 yearsThanks @Martin Brandenburg :))
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Marc van Leeuwen about 10 yearsIs there any reason to talk about subspaces instead of subsets here? -
Martin Brandenburg about 10 yearsYes, 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).
