Every Hausdorff topological space is sober, hence can be identified with a spatial locale; furthermore, every locally compact Hausdorff topological space is also Hausdorff as a locale. On the other hand, there seems to be no reason that a Hausdorff locale should have any points at all, let alone be spatial. What are some interesting examples of non-spatial Hausdorff locales?
(A Hausdorff locale is a locale $X$ such that the diagonal $\Delta : X \to X \times X$ is a closed embedding.)
Since every sublocale of a Hausdorff locale is Hausdorff, in principle it suffices to find a non-spatial sublocale of any Hausdorff locale. For example, if $K$ is an compact Hausdorff space with no isolated points, then the smallest dense sublocale of $K$ is Hausdorff but has no points. However, I am more interested in "natural" examples, especially ones where Hausdorff-ness has some extrinsic interpretation related to the construction of the example.
