If all compact subspaces of $X$ are closed in $X$, then is $X$ Hausdorff ? If not then please give counter-example.
We know that converse is true i.e. compact subspace of Hausdorff space is closed.
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Shak
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2They are called $T_B$ spaces or $KC$ spaces. – Daniel Fischer Jun 12 '18 at 13:52
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@DanielFischer I feel they need not be Hausdorff, but I have not been able to produce any counterexample. – Shak Jun 12 '18 at 13:56
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This may help. I think constructing a non-Hausdorff example is nontrivial. – Daniel Fischer Jun 12 '18 at 14:09
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2Actually, it's not so hard. – Daniel Fischer Jun 12 '18 at 14:11