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A topological space $X$ is called locally Hausdorff if every point has a Hausdorff neighborhood.

Local Hausdorffness is an interesting separation axiom that is strictly weaker than Hausdorff, but strictly stronger than $T_1$: The line with doubled origin is locally Hausdorff but not Hausdorff, and an infinite set with the cofinite topology is $T_1$ but not locally Hausdorff. On the other hand, it is not hard to see that a locally Hausdorff space is $T_1$.

However, I have not been able to find any good references on properties of locally Hausdorff spaces. For example, I am interested in how various definitions of local compactness, which coincide for Hausdorff spaces, interact with the weaker axiom of local Hausdorffness.

Does anybody know of a collection of facts about local Hausdorffness, major theorems, book chapters, articles, etc.?

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  • $\begingroup$ Compactness need not imply local compactness in a locally Hausdorff space. I once asked about this, see here $\endgroup$ – Stefan Hamcke Dec 2 '15 at 0:09
  • $\begingroup$ @StefanHamcke I had always thought that local compactness should mean that every point has a compact neighborhood, which is equivalent, in a Hausdorff space, to each point having a neighborhood base of compacts. There is also the requirement that each point has a closed compact neighborhood, which should be stronger in the locally Hausdorff case, but still weaker than compactness. $\endgroup$ – Slade Dec 2 '15 at 3:16
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S.B. Niefeld, A note on the locally Hausdorff property, Cahiers de Topologie et Géométrie Différentielle Catégoriques, $24$ no. $1$ $(1983)$, p. $87$-$95$, has some results. Much of the paper is category-theoretic, but there are some purely topological results.

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  • $\begingroup$ This is exactly what I was looking for. Lots of very relevant things to chew on. Thanks! $\endgroup$ – Slade Dec 2 '15 at 10:57
  • $\begingroup$ @Slade: You’re welcome! I’m glad that it helped; I wasn’t sure that there was enough there to be useful. $\endgroup$ – Brian M. Scott Dec 2 '15 at 11:13
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    $\begingroup$ Well, it just so happens that it intersects many of the ideas that led me to post this question in the first place, like stable compactness, locales, and some other things. For example, I think the spaces I'm working with should be strongly locally Hausdorff. $\endgroup$ – Slade Dec 2 '15 at 11:21

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