### Decomposing Baire class 1 functions into continuous functions

by Shelah and Steprans. [ShSr:510]

Fundamenta Math, 1994

Let 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) <= dec <= 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 d is denoted the
least cardinal of a dominating family in omega^omega . Steprans
showed that it is consistent that cov(Meager) not= dec . In
this paper we show that the second inequality can also be made
strict. The model where dec is different from 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)
not= dec, is a slight modification of the
iteration of super-perfect forcing.

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