σ-Continuous and Co-σ-continuous Maps

Abstract

In this chapter we isolate the topological setting that is suitable for our study. We first present 2.1–2.3 to follow an understandable logical scheme nevertheless the main contribution are presented in 2.4–2.7 and our main tool will be Theorem 2.32. An important concept will be the σ-continuity of a map Φ from a topological space (X, T) into a metric space (Y, g). The σ-continuity property is an extension of continuity suitable to deal with countable decompositions of the domain space X as well as with pointwise cluster points of sequences of functions Φn: X → Y, n = 1,2,… When (X,T) is a subset of a locally convex linear topological space we shall refine our study to deal with σ-slicely continuous maps, the main object of these notes. When (X, T) is a metric space too we shall deal with σ-continuity properties of the inverse map Φ_1that we have called co-σ-continuity.