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In a recent pull request to the π-Base, it was observed that all locally compact and KC (Kompacts are Closed) spaces are regular: since compacts are closed, each local neighborhood base of compacts is a local neighborhood base of closed sets, which is equivalent to regular. See also this MathSE post: Show that every locally compact Hausdorff space is regular.

A natural weakening of KC is weakly Hausdorff: for every compact Hausdroff space $K$ and continuous map $f:K\to X$, the image $f[K]$ is closed in $X$.

Are all locally compact and weakly Hausdorff spaces regular?

Steven Clontz
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    Since weakly Hausdorff spaces are $T_1$, $T_1$ regular spaces are Hausdorff, and locally compact Hausdorff spaces are regular, the question is the same as asking whether all locally compact weakly Hausdorff spaces are Hausdorff. – David Gao Jul 01 '24 at 08:45
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    Any locally compact KC space is Hausdorff and hence regular and compactly generated. On the other hand, any compactly generated weak Hausdorff space is KC. So basically the question is, is there a locally compact, weak Hausdorff space which is not compactly generated? There are locally compact $T_1$ spaces which are not compactly generated, but I don't know if there are weak Hausdorff examples. – Tyrone Jul 02 '24 at 06:21
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    Tyrone, any hints on the $T_1$ example that I could contribute to $\pi$-Base? Looks like we have a gap there. – Steven Clontz Jul 04 '24 at 23:13
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    Hi Steven. The example I know is due to J. Isbell. I wrote about it in the answer here. Isbell exaplins in the linked paper how to convert the discussed space into a $T_1$ space with the same properties. (Forgive the poor writing in my post - it is a few years old by now...) – Tyrone Jul 09 '24 at 12:49

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