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

Taras Banakh


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 $f{\restriction}X_n$ is continuous. By $\sigma$ (resp. $\bar\sigma$) 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 $\omega_1\le\mathfrak_{\bar\sigma}\le\mathfrak c_\sigma\le\mathfrak c$. We prove that $\mathfrak p\le\mathfrak q_0=\mathfrak c_{\bar\sigma}=\min\{\mathfrak c_\sigma,\mathfrak b,\mathfrak q\}\le\mathfrak c_\sigma\le\min\{non(\mathcal M),non(\mathcal N)\}$.

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DOI: https://doi.org/10.2478/tmmp-2020-0014