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We Know that all closed subsets of a compact topological space $(X,\tau)$ are compact. But if we add The Hausdorff condition on the topology $\tau$ we could see the equivalence of these subsets.(i.e. in compact Hausdourff spaces closed subsets are the same as compact subsets)

Know for asking the converse of the above fact we could or not omit the compactness of the space$(X,\tau)$ as follows:

  • (STATEMENT) If all compact subsets of a topological space $(X,\tau)$ are closed then $(X,\tau)$ is Hausdorff.

  • If the above statement is not valid, Is there a separation axiom weaker than Hausdorffness on the space $X$ that compact subsets are closed?

For the first statement If we add the condition of compactness of $(X,\tau)$, it changes as follows:

  • Is The space $(X,\tau)$ Hausdorff,If closed subsets and compact subsets are equivalent in $X$?
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It may be valuable to look up the notion of weak hausdorff which appears frequently in homotopy theory. –  David White Jun 1 '12 at 20:06
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And, here's an example that STATEMENT is false: mathoverflow.net/questions/88420/… –  David White Jun 1 '12 at 20:08
    
Thank you very much dear White. But You only show to me that The first statement is false. But as you have seen, the next question asked that for which category of spaces, compact subsets are closed? I didn't see anything about Asking for it.(best wishes) –  Ali Reza Jun 1 '12 at 20:20
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Some people consider Weak Hausdorff to be a separation axiom (weaker than Hausdorff, of course). See for instance: mathoverflow.net/questions/86812/separation-axioms. That post claims that Weak Hausdorff implies T1, so if you are only interested in T0-T6 then I guess T1 should do the job. Also, nLab discusses in some detail how Weak Hausdorff implies T0: ncatlab.org/nlab/show/compactly+generated+topological+space –  David White Jun 1 '12 at 20:40
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Spaces where compact sets are closed are called KC-spaces. Here is a monthly article which discusses the topologies between $T_1$ and $T_2$. jstor.org/stable/2316017 –  Gjergji Zaimi Jun 1 '12 at 21:00
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1 Answer

The Statement in the question is false. A counter-example can be found in this MO answer. As for a separation axiom weaker than Hausdorff which make compact subsets closed, one such notion is that of a Weak Hausdorff space.

As Gjergji points out, spaces where compact subsets are closed are called KC-spaces. Hausdorff implies KC, but not conversely (this answers the OP's third question).

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I just wanted to turn my comments into an answer, so this wouldn't come back to the front-page. I should have done that in the first place, way back when this was first asked. –  David White Jan 12 '13 at 18:57
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