Quasi-continuous selections

Abstract

In this paper we will study the existence of a quasi-continuous selection for a multifunction F: X → Y which is upper Baire continuous with respect to compact complement topology. The upper Baire continuity means that for any open set V the upper inverse image F+(V ) = (x: F(x) ⊂ V ) is of the form (G \ S) ∪ T, where G is of second category and open, S; T are of first category and T is a subset of the closure of G. This type of continuity seems to be very close to the Baire property of mappings, and the upper Baire continuity has three nice features.