Compact space and Hausdorff space


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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Updated on August 01, 2022


  • Aera
    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.

  • Aera
    Aera about 10 years
    Thanks @Martin Brandenburg :))
  • Marc van Leeuwen
    Marc van Leeuwen about 10 years
    Is there any reason to talk about subspaces instead of subsets here?
  • Martin Brandenburg
    Martin Brandenburg about 10 years
    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).