Extending Baire-one functions on topological spaces
The paper consists of six sections: Introduction, Preliminaries, Extension of Baire-one functions, Counterexamples and questions, Extension of mappings of the first Borel class, Extension of Baire-one functions on compact convex sets. In the introduction the authors describe the history of the problem, the main result and give the summary of two last sections. Section 2 contains necessary definitions and propositions. In Section 3 the authors prove their main result on extending Baire-one functions (Theorem 13): Let \(Y\) be a Lindelöf hereditarily Baire subset of a completely regular space \(X\) and \(f\) be a Baire-one function on \(Y\). Then there exists a Baire-one function \(g\) on \(X\) such that \(f=g\) on \(Y\), \(\inf (f(Y)_=\inf (g(X))\), \(\sup (f(Y))=\sup(g(X))\). In Section 4 the authors collect several examples showing that the assumptions of their main theorem cannot be weakened in a natural way. They also collect here some open questions. In Section 5 the authors show that once it is possible to extend Baire-one functions, the extension theorem can be obtained even for \(F_\sigma\)-measurable mappings with values in Polish spaces. The last section of the paper is devoted to an application in convex analysis (which motivated the authors research).
scientific article; zbMATH DE number 2173004