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Decomposing Baire class 1 functions into continuous functions. (English) Zbl 0821.03022

It is shown that it is relatively consistent with ZFC that every Baire class 1 function is the union of \(\omega_ 1\) continuous functions and the minimum cardinality of a dominating family in \(\omega^ \omega\) is \(\omega_ 2\). The model is obtained by the \(\omega_ 2\)-iteration of superperfect forcing with countable support over a model of CH.
The cardinal \({\mathfrak {dec}}\) stands for the smallest cardinal \(\kappa\) such that every Baire class 1 function is the union of \(\kappa\) continuous functions. The cardinal \({\mathfrak d}\) is the minimum cardinality of a dominating family in \(\omega^ \omega\). The cardinal \(\text{cov} (\mathbb{K})\) is the minimal cardinality of a covering of the real line with nowhere dense sets. J. Cichoń, M. Morayne, J. Pawlikowski, and S. Solecki [J. Symb. Logic 56, 1273-1283 (1991; Zbl 0742.04003)] have shown that: \(\text{cov} (\mathbb{K})\leq {\mathfrak {dec}}\leq {\mathfrak d}\) and they asked if these inequalities can be made strict. J. Steprāns [ibid. 58, 1268-1283 (1993; Zbl 0805.03036)] has shown relatively consistent with ZFC that \(\text{cov} (\mathbb{K})< {\mathfrak {dec}}\). The present paper shows that the second inequality \({\mathfrak {dec}}< {\mathfrak d}\) can be made strict.

MSC:

03E15 Descriptive set theory
03E35 Consistency and independence results
26A21 Classification of real functions; Baire classification of sets and functions
03E50 Continuum hypothesis and Martin’s axiom
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