A New Diagonal Separation and its Relations With the Hausdorff Property

Let P be a property of subobjects relevant in a category C. An object X∈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 paper we study the locales (frames) in which the diagonal is fitted (i.e., an intersection of open sublocales—we speak about F -separated locales). Recall that a locale is fit if each of its sublocales is fitted. Since this property is inherited by products and sublocales, fitness implies (F sep) which is shown to be strictly weaker (one of the results of this paper). We show that (F sep) is in a parallel with the strong Hausdorff axiom (sH): (1) it is characterized by a Dowker-Strauss type property of the combinatorial structure of the systems of frame homomorphisms L→M (and therefore, in particular, it implies (T_U) for analogous reasons like (sH) does), and (2) in a certain duality with (sH) it is characterized in L by all almost homomorphisms (frame homomorphisms with slightly relaxed join-requirement) L→M being frame homomorphisms (while one has such a characteristic of (sH) with weak homomorphisms, where meet-requirement is relaxed).