Decomposing Baire class 1 functions into continuous functions

TL;DR

This paper investigates the decomposition of Baire class 1 functions into continuous functions, establishing the consistency of strict inequalities between certain cardinal invariants related to these functions.

Contribution

It demonstrates that the inequality dec <= d can be strict, complementing previous results on cov(Meager) and dec, using models with added Miller reals.

Findings

dec can be strictly less than d in certain models

The model involves adding omega_2 Miller reals to a CH universe

The inequalities between cardinal invariants can be strict in different models

Abstract

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.