nLab
weakly Hausdorff topological space

Definition

A topological space $X$ is weakly Hausdorff (or weak Hausdorff) if for any compact Hausdorff space $K$ and every continuous map $f:K\to X$, the image $f(K)$ is closed. Every weakly Hausdorff space is $T_1$ (that is every point is closed), and every Hausdorff space is weakly Hausdorff. For the most common purposes for which Hausdorff spaces are used, the assumption of being weakly Hausdorff suffices. See also compactly generated space.

We have given the definition for topological spaces, but it also makes sense as stated for locales. Where these overlap (sober spaces and topological locales), they agree given the ultrafilter theorem (which implies that all compact Hausdorff spaces/locales are sober/topological).

Weak Hausdorffification

(this is a left adjoint …)