I know that in a Hausdorff space, every compact set is closed.

However, is it true that if every compact set is closed, then the space is necessarily Hausdorff?

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    $\begingroup$ I suspect that you meant to ask the question in the title, not the one in the body of the question. $\endgroup$ – Brian M. Scott Mar 12 '13 at 19:26
  • $\begingroup$ The title and the body ask different questions. $\endgroup$ – Asaf Karagila Mar 12 '13 at 19:26
  • $\begingroup$ The title and question are different. The title asks if every compact set is also a closed set, is the space Hausdorff. The question asks if every set is closed and compact, is it Hausdorff. Which do you mean? $\endgroup$ – anonymous Mar 12 '13 at 19:26
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    $\begingroup$ Assuming the version in the title, such a space is at least $T_1$ because points are closed. $\endgroup$ – JSchlather Mar 12 '13 at 19:30

I’m answering the question in the title. Let $X$ be an uncountable set, and let $\tau$ be the co-countable topology on $X$. The compact sets in $\langle X,\tau\rangle$ are precisely the finite sets, which are all closed, but $X$ is not Hausdorff.

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    $\begingroup$ yeah that would be mine too, here you see that the uniqueness of limits of sequences does not imply hausdorff $\endgroup$ – Dominic Michaelis Mar 12 '13 at 19:33
  • $\begingroup$ @Dominic: Sequences are worthless without assuming first countability. $\endgroup$ – Asaf Karagila Mar 12 '13 at 19:41
  • $\begingroup$ @Matt: Look again: the topology is the co-countable topology, not the cofinite topology. $\endgroup$ – Brian M. Scott Jan 6 '15 at 22:09

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