On Baire classification of mappings with values in connected spaces

Abstract

We generalize the Lebesgue-Hausdorff Theorem on the characterization of Baire-one functions for σ-strongly functionally discrete mappings defined on arbitrary topological spaces.1991 Mathematics Subject Classification. Primary 54C08, 26A21; Secondary 54C50, 54H05.

On composition of Baire functions

“…For α = 0 the statement is evident. 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.…”

The first homotopic Baire class of maps with values in ANR's coincides with the first Baire class

“…Let ϕ : Z → Y be a weak local homeomorphism. According to Theorem 3.1 there exists a map g ∈ Σ s 1 (X, Z) such that f = ϕ • g. Since Z is path-connected and locally path-connected, [9,Theorem 4.1] implies that g ∈ B 1 (X, Z). Furthermore, as Z is contractible, we get from (1) of Proposition 2.2 that g ∈ hB 1 (X, Z).…”

Extensions of vector-valued functions with preservation of derivatives

“…Now, it is easy to deduce that L is F σ -measurable on F and [KZ] deduce that L is Baire one. For this step with vector-valued functions, we need to refer to [Ka,Corollary 4.13]. Alternatively, we construct a sequence of continuous functions {L m } m∈N that converges point-wise to L. Observe that F m+1 ⊃ F m for every m ∈ N and let H n (n ∈ N) be an increasing sequence of finite sets such that F \ der F ⊂ m∈N H m ⊂ F. By Dugundji's extension theorem or our Theorem 1.1(ii)(v), 5 we let L m : F → L(R n , Y) be a continuous extension of the (continuous) function L| F m ∪H m .…”