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Let $X$ be topological space such that every its closed subset has finitely many connected componenets. Is there any charactrization for such topological space?

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  • $\begingroup$ Please edit to correct to "closed subsets" and "finitely" (each both in the title and text) and also correct to "components" in the title and "characterization" in the text and "spaces" for the last word (also "every its closed subset" should be "all its closed subsets"). $\endgroup$
    – YCor
    Nov 3, 2015 at 8:04
  • $\begingroup$ Please edit once more to get "components" in both the title and text and "characterization" in the text, and change "such that every its closed subset has" to either "such that all its closed subsets have" or "in which every closed subset has". $\endgroup$
    – YCor
    Nov 3, 2015 at 10:26
  • $\begingroup$ If $X$ is Hausdorff, it must be finite. The non-Hausdorff case seems difficult to completely characterize. $\endgroup$ Nov 3, 2015 at 15:55

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