Separation of finely closed sets by finely open sets

Abstract

It is well known that the fine topology in potential theory and the density topology in real analysis are not normal. This means that there exist pairs of disjoint finely (density) closed sets which cannot be separated by disjoint finely (density) open sets. A natural question arises about which pairs can be separated. We study those pairs of disjoint finely (density) closed sets which can be separated by disjoint finely (density) open sets. The key tool is the Lusin-Menchoff property of fine (density) topology. The main result is that finely (density) closed sets are finely (density) separated iff they are \(F_{\sigma }\)-“semiseparated” (Theorem 2.1, Theorem 2.2).