A new diagonal separation and its relations with the Hausdorff property (Q2121602)

From the facts that the Hausdorff property of a topological space \(X\) is characterized by the closedness of the diagonal in \(X \times X\) and the strong Hausdorff axiom in the category of locales (introduced by \textit{J. R. Isbell} [Math. Scand. 31, 5--32 (1972; Zbl 0246.54028)]), the authors introduce the following interesting and useful criterion for separation: ``Let \(\mathcal P\) be a property of subobjects relevant in a category \(C\). An object \(X \in C\) is \(\mathcal P\)-separated if the diagonal in \(X \times X\) has \(\mathcal P\)''. Using this idea, the authors study the locales (frames) in which the diagonal is fitted (i.e., an intersection of open sublocales -- they speak about \(\mathcal F\)-separated locales). Since the intersection of open sublocales is an operation of closure type, it is natural to ask about fitted diagonals; i,e, the property \(\mathbf{\mathcal{F}sep}\) or of the \(\mathcal{F}\)-separated locales (frames). The study of this property is the main topic of this paper. On the other hand, taking into account that a well-known property of a locale (frame) is fitness [\textit{J. R. Isbell}, Math. Scand. 31, 5--32 (1972; Zbl 0246.54028)], that a locale is \textit{fit} if each of its sublocales is fitted, and that fitness is preserved under products and sublocales, the authors make an immediate observation that \textit{fit} implies \(\mathbf{\mathcal{F}sep}\). Hence the first question one may ask is whether this implication can be reversed (it cannot; \(\mathbf{\mathcal{F}sep}\) is strictly weaker than fitness, which is one of the results of this paper). Later, the authors prove the Dowker-Strauss type characterization of the strong Hausdorff property, and Finally, they prove that \(\mathbf{\mathcal{F}sep}\) is strictly weaker than fitness, and that it does not coincide with any of the three standard axioms weaker than fitness: subfitness, weak subfitness and prefitness.