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.?
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.