I know that every Hausdorff space $X$ has the following property:
$A\subset X$ is compact $\Rightarrow A$ is closed
In other words, that is necessary for $X$ to be Hausdorff. But is it sufficient? If not, what information can get from this property?
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2It isn't – Daniel Fischer Nov 01 '16 at 21:42
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1A standard example of a $KC$-space (compact sets are closed) that is not Hausdorff is $\Bbb Q^*$, the one-point compactification of the rationals. – Brian M. Scott Nov 02 '16 at 00:23
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1I wonder what can we conclude from this $KC$-property. If it doesn't imply $X$ is $T_2$, does it imply $T_0$ or $T_1$? Or what additional property(ies) would guarantee $X$ is Hausdorff (or $T_0, T_1$ etc)? – rmdmc89 Nov 02 '16 at 02:23