Making Borel functions look continuous. (English) Zbl 0964.03051

Let be \(g:X\to {\mathbb R}\) where \(X\) is a separable metric space. The function \(g\) is called a connectivity function if the graph of the restriction of \(g\) to any connected subset of \(X\) is a connected subset of \(X\times {\mathbb R}\); \(g\) is called an extendable connectivity function if there exists a connectivity function \(G:X\times [0,1]\to {\mathbb R}\) such that \(g(x)=G(x,0)\) for any \(x\in X\).
Let \({\mathbf B}_\alpha \) denote the set of all functions \(f:{\mathbb R}\to {\mathbb R}\) for which preimages of open sets are in the Borel class \(\Sigma ^0_{\alpha +1}\) if \(\alpha <\omega \), and in the Borel class \(\Sigma ^0_\alpha \) if \(\omega \leq \alpha <\omega _1\) (i.e. the functions of the Baire class \(\alpha \) coincide with \({\mathbf B}_\alpha \) if \(\alpha <\omega \) and with \({\mathbf B}_{\alpha +1}\) if \(\alpha \geq \omega \)). The following result is proved: For any \(\alpha \), \(1\leq \alpha <\omega _1\), there exists a function \(f\) in \({\mathbf B}_\alpha \) such that for any function \(g\) in \(\bigcup _{\gamma <\alpha}{\mathbf B_\gamma}\), \(f+g\) is an extendable connectivity function. This is a generalization of two results obtained by Natkaniec and Recław.


03E15 Descriptive set theory
26A15 Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable
26A21 Classification of real functions; Baire classification of sets and functions