Functionally σ-discrete mappings and a generalization of Banach's theorem

Abstract

We present σ-strongly functionally discrete mappings which expand the class of σ-discrete mappings and generalize Banach's theorem on analytically representable functions.

On composition of Baire functions

“…The equality (i) for α = 1 was proved in [12,Theorem 5]. This fact and [11,Theorem 22] imply the equalities (i) and (ii) for all α > 1.…”

“…If a map f has a σ-(strongly functionally) discrete base which consists of (functionally) ambiguous sets of the α'th class, then we say that f belongs to the class Σ α (X, Y ) (or to Σ f α (X, Y ), respectively). We will use the next result which, in fact, was established in [11] and [12]. Theorem 1.…”

On Baire-one mappings with zero-dimensional domains

“…Then B k is a σ-sfd family and X = ∪B k for every k. According to [16,Lemma 13] for every k ∈ N there exists a sequence (B k,n ) ∞ n=1 of sfd families of zero subsets of X such that B k,n ≺ B k , B k,n ≺ B k,n+1 for every n ∈ N and ∞ n=1 B k,n = X. For all k, n ∈ N we set…”

“…In this paper we develop technics from [6] and generalize the Lebesgue-Hausdorff Theorem for σ-discrete mappings defined on strongly zero-dimensional spaces with valued in metrizable spaces. In order to do this we consider the class of σ-strongly functionally discrete mappings introduced in [16]. We denote this class by Σ f (X, Y ) and notice that Σ f (X,…”

On Baire classification of mappings with values in connected spaces

“…Then B k is a σ-sfd family and X = ∪B k for every k. According to [9,Lemma 13] for every k ∈ N there exists a sequence (B k,n ) ∞ n=1 of sfd families B k,n = (B k,n,i : i ∈ I k,n ) of functionally closed subsets of X such that B k,n ≺ B k , B k,n ≺ B k,n+1 for every n ∈ N and…”