Selection principle \(\mathrm{S}_{1}\) and combinatorics of open covers
Some relationships between covering properties of a topological space $X$ and properties of the space of real continuous functions $C_p(X)$ are studied. If $\mathcal{A}, \mathcal{B}\subseteq\mathcal{P}(Y)$ are sets of subsets of a set $Y$, then $\mathrm{S}_1(\mathcal{A}, \mathcal{B})$ means the following: for any sequence $\{\mathcal{U}_n\}_{n\in\omega}$ of elements of $\mathcal{A}$, for every $n\in\omega$ there exists an $U_n\in\mathcal{U}_n$ such that $\{U_n: n\in\omega\}\in\mathcal{B}$. In this paper the author obtains some properties of topological spaces of real functions on $X$ equivalent to the property $\mathrm{S}_1(\mathcal{A}, \mathcal{B})$ or $\mathrm{S}_1(\mathcal{A}^{sh}, \mathcal{B})$ for $\mathcal{A}, \mathcal{B}$ being the family of all open, open $\omega$- or open $\gamma$-covers of a topological space $X$, where $\mathcal{A}^{sh}$ denotes the families of shrinkable covers of a cover $\mathcal{A}$. For example, the following is proved: Assume that $\Phi$ is one of the symbols $\Omega$ ($\omega$-covers) and $\Gamma$ ($\gamma$-covers), and $\Psi$ is one of the symbols $\mathcal{O}$ (open covers), $\Omega$ and $\Gamma$. Then for any couple $\langle\Phi, \Psi\rangle$ different from $\langle\Omega, \mathcal{O}\rangle$, a topological space $X$ is an $\mathrm{S}_1(\Phi, \Psi)$-space if and only if USC$_p(X)^+$ satisfies the selection principle $\mathrm{S}_1(\Phi_0, \Psi_0)$.
scientific article; zbMATH DE number 7047521