Let P be a property of subobjects relevant in a category C. An object X in C is P-separated if the diagonal in X × X has P; thus e.g. closedness in the category of topological spaces (resp. locales) induces the Hausdorff (resp. strong Hausdorff) axiom.
In this talk we consider the locales in which the diagonal is fitted (i.e., an intersection of open sublocales - we speak about F-separated locales).
Among others, we will show that F-separatedness is a property strictly weaker than fitness. Moreover, we will explore a surprising parallel with the strong Hausdorff axiom, including a Dowker-Strauss type theorem and a characterization in terms of certain relaxed morphisms.

This is a joint work with Jorge Picado and Ales Pultr.