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This is something of a converse to the theorem which says if $X$ is Hausdorff and $C$ is a compact subset of $X$, then $C$ is closed in $X$. But I do not know if true or not.

Question: If $X$ is T$_1$ but not Hausdorff, then does $X$ necessarily have a compact subset that is not closed?

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    no edit needed ;) – 2017-01-15
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    Relevant: http://math.stackexchange.com/questions/955382/spaces-where-all-compact-subsets-are-closed – 2017-01-15
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    @JoeJohnson126 ooh thank you. in my head I was sort of thinking that $X$ is T$_1$. If single points are not closed then the space is not very interesting to me. – 2017-01-15
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    @ForeverMozart, then add that condition to the body of the question. – 2017-01-15
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    Related: https://math.stackexchange.com/questions/328725 – 2017-01-15

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