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This question already has an answer here:

One example of a non-Hausdorff topological space in which all compact subsets are closed is the co-countable topology on an uncountable set, as demonstrated here.

It was claimed (as a now-deleted answer to the above question) that the compact complement topology on $\mathbb{R}$ was another example, but this was proven incorrect here.

Can someone provide an additional example of a non-Hausdorff space in whihc all compact subsets are closed?

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marked as duplicate by Brian M. Scott general-topology Sep 1 '15 at 19:02

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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    $\begingroup$ You are looking for a topological space. Can you make precise which properties this space should have and which I must not have? $\endgroup$ – principal-ideal-domain Sep 1 '15 at 11:59
  • $\begingroup$ The linked question has two answers; one is the space that you already have, but the other is different. $\endgroup$ – Brian M. Scott Sep 1 '15 at 19:03

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