$\sigma$-Continuous functions and related cardinal characteristics of the continuum

Abstract

A function $f:X\to Y$ between topological spaces is called $\sigma$-$continuous$ (resp. $\bar{\sigma }$-$continuous$) if there exists a (closed) cover $\{X_n{\}}_{n\in \omega }$ of $X$ such that for every $n\in \omega$ the restriction $f\upharpoonright X_n$ is continuous. By ${\mathfrak{c}}_{\sigma }$ (resp. ${\mathfrak{c}}_{\bar{\sigma }}$) we denote the largest cardinal $\kappa \le \mathfrak{c}$ such that every function $f:X\to \mathbb{R}$ defined on a subset $X\subset \mathbb{R}$ of cardinality $|X|<\kappa$ is $\sigma$-continuous (resp. $\bar{\sigma }$-continuous). It is clear that ${\omega }_1\le {\mathfrak{c}}_{\bar{\sigma }}\le {\mathfrak{c}}_{\sigma }\le \mathfrak{c}$. We prove that $\mathfrak{p}\le {\mathfrak{q}}_0={\mathfrak{c}}_{\bar{\sigma }}=\min \{{\mathfrak{c}}_{\sigma },\mathfrak{b},\mathfrak{q}\}\le {\mathfrak{c}}_{\sigma }\le \min \{\mathrm{non}(\mathcal{M}),\mathrm{non}(\mathcal{N})\}$. The equality ${\mathfrak{c}}_{\bar{\sigma }}={\mathfrak{q}}_0$ resolves a problem from the initial version of the paper.

7 pages

References in corpus (1)

• On critical cardinalities related to $Q$-sets