# Sh:510

• Shelah, S., & Steprāns, J. (1994). Decomposing Baire class 1 functions into continuous functions. Fund. Math., 145(2), 171–180.
• Abstract:
Let \mathfrak{dec} be the least cardinal \kappa such that every function of first Baire class can be decomposed into \kappa continuous functions. Cichon, Morayne, Pawlikowski and Solecki proved that cov(Meager)\leq \mathfrak{dec}\leq \mathfrak{d} and asked whether these inequalities could, consistently, be strict. By cov(Meager) is meant the least number of closed nowhere dense sets required to cover the real line and by \mathfrak{d} is denoted the least cardinal of a dominating family in \omega^\omega. Steprans showed that it is consistent that cov(Meager)\neq \mathfrak{dec}. In this paper we show that the second inequality can also be made strict. The model where \mathfrak{dec} is different from \mathfrak{d} is the one obtained by adding \omega_2 Miller - sometimes known as super-perfect or rational-perfect - reals to a model of the Continuum Hypothesis. It is somewhat surprising that the model used to establish the consistency of the other inequality, cov(Meager)\neq\mathfrak{dec}, is a slight modification of the iteration of super-perfect forcing.
• Version 1994-01-13_10 (8p) published version (10p)
Bib entry
@article{Sh:510,
author = {Shelah, Saharon and Stepr{\={a}}ns, Juris},
title = {{Decomposing Baire class $1$ functions into continuous functions}},
journal = {Fund. Math.},
fjournal = {Fundamenta Mathematicae},
volume = {145},
number = {2},
year = {1994},
pages = {171--180},
issn = {0016-2736},
mrnumber = {1297403},
mrclass = {03E35 (03E05 03E15 26A21)},
note = {\href{https://arxiv.org/abs/math/9401218}{arXiv: math/9401218}},
arxiv_number = {math/9401218}
}