Compositions of Sierpiński-Zygmund functions from the left (Q2747069)

A function \(f: {\mathbb R}\to {\mathbb R}\) is a Sierpiński-Zygmund function (\(f\in\) \text{SZ}) if the restriction \(f|A\) is continuous for no \(A\subset{\mathbb R}\) with \(|A|={\mathbf c}\). \(\mathcal R\) is the class of all functions \(f:{\mathbb R}\to {\mathbb R}\) with \(|f^{-1}(y)|<{\mathbf c}\) for every \(y\in {\mathbb R}\). The cardinal \(c_l(\text{SZ})\) is defined as the minimal cardinality of a family \({\mathcal F}\subset {\mathcal R}\) such that there is no \(h\in \text{SZ}\) with \(f=h\circ g\) for every \(f\in {\mathcal F}\) and some \(g\in \text{SZ}\). (\(c_l(\text{SZ})=(2^{\mathbf c})^+\) if such ``universal'' \(h\) exists for the family \(\mathcal F\) of all functions from \(\mathbb R\) to \(\mathbb R\).) The cardinal function \(c_l\) has been introduced by \textit{T. Natkaniec} [Bull. Pol. Acad. Sci., Math. 44, No. 2, 251-256 (1995; Zbl 0876.26004)]. The cardinal \(c_l(\text{SZ})\) has been considered by \textit{K. Ciesielski} and \textit{T. Natkaniec} [Topology Appl. 79, 75-99 (1997; Zbl 0890.26002)]. They proved that if the covering of category is \(\mathbf c\) and \(\mathbf c\) is a regular cardinal then \({\mathbf c}<c_l(\text{SZ})\). In this paper the author proves the following. (1) If \({\mathbf c}=\omega_1\) then \(c_l(\text{SZ})=b_c\). (\(b_c\) is the bounding number of the continuum, i.e., the minimal cardinality of a bounding family of functions from \({\mathbb R}\) to \({\mathbb R}\), i.e., such family \(\mathcal F\) that for any \(g: {\mathbb R}\to {\mathbb R}\) there is \(f\in {\mathcal F}\) with \(|[f\geq g]|={\mathbf c}\).) Hence \(c_l(\text{SZ})\leq 2^{\mathbf c}\) and \(c_l(\text{SZ})\) may be any regular cardinal between \({\mathbf c}\) and \(2^{\mathbf c}\). (2) If the covering of category is \(\mathbf c\) and \(\mathbf c\) is a limit cardinal then \(c_l(\text{SZ})=(2^{\mathbf c})^+\). This solves a question of Ciesielski and Natkaniec.