Shelah, Saharon; Steprāns, Juris Decomposing Baire class 1 functions into continuous functions. (English) Zbl 0821.03022 Fundam. Math. 145, No. 2, 171-180 (1994). 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. Reviewer: A.W.Miller (Madison) Cited in 2 Documents 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 Keywords:Baire class 1 function; superperfect forcing Citations:Zbl 0742.04003; Zbl 0805.03036 PDFBibTeX XMLCite \textit{S. Shelah} and \textit{J. Steprāns}, Fundam. Math. 145, No. 2, 171--180 (1994; Zbl 0821.03022) Full Text: arXiv EuDML