# General facts about locally Hausdorff spaces?

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.?

• Compactness need not imply local compactness in a locally Hausdorff space. I once asked about this, see here – Stefan Hamcke Dec 2 '15 at 0:09
• @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. – Slade Dec 2 '15 at 3:16

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.