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Is there a name for a topological space $X$ in which every proper closed subset is compact$^{(*)}$? It is well known that in a compact topological space, every closed set is compact. Hence, the class of compact spaces is contained in the class of spaces with $(*)$ property.

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marked as duplicate by user99914, Noah Schweber, Martin Sleziak, Henno Brandsma general-topology Jun 14 '16 at 6:03

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  • $\begingroup$ The total space $X$ should be closed in any topology, so what you find is just a compact space. $\endgroup$ – cjackal Jun 14 '16 at 5:13
  • $\begingroup$ Thx for this note, i fixed it. $\endgroup$ – M.A. Jun 14 '16 at 5:18
  • $\begingroup$ Do you have an interesting example of such a space? $\endgroup$ – Hoot Jun 14 '16 at 5:20
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    $\begingroup$ I believe it still has to be compact. Otherwise you can find a proper sequence in the space with no accumulation point. $\endgroup$ – Forever Mozart Jun 14 '16 at 5:20
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    $\begingroup$ math.stackexchange.com/questions/1317724/… $\endgroup$ – John M Jun 14 '16 at 5:21