Bounded resolutions for spaces \(C_p(X)\) and a characterization in terms of \(X\)
The authors study topological spaces \(X\) with the property of \textit{nice framing}. Recall that a family \(\{ U_{\alpha,n}\colon \alpha\in\omega^\omega, n\in\omega\}\) of closed subsets of \(X\) is called a \textit{framing} if: (i) for each \(\alpha\in\omega^\omega\) the layer \(\{ U_{\alpha,n}\colon n\in\omega\}\) is an increasing covering of \(X\), and (ii) for every \(n\in\omega\) one has that \(U_{\beta,n}\subset U_{\alpha,n}\), \(\alpha\le\beta\), see [\textit{J. C. Ferrando}, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 114, No. 2, Paper No. 107, 34 p. (2020; Zbl 1436.54012)]. \(X\) has a \textit{nice framing} if it admits a framing \(\{ U_{\alpha,n}\colon \alpha\in\omega^\omega, n\in\omega\}\) in \(X\) enjoying the property: for every \(f \in C(X)\) there exists \(\alpha\in\omega^\omega\) such that \(| f (x)| \le n\) for each \(x\in U_{\alpha,n}\) and \(n\in\omega\). Let \(X\) be a Tychonoff space. The space \(C_p(X)\) is \textit{ \(K\)-analytic-framed in \(\mathbb{R}^X\)} if there exists a \(K\)-analytic space \( Z\) such that \(C_p(X) \subset Z \subset\mathbb{R}^X\). A Tychonoff space \(X\) admits a \textit{fundamental resolution of functions} if there a family \(\{ f_\alpha\colon\alpha\in\omega^\omega\}\) of nonnegative real-valued functions on \(X\) such that \(f_\alpha\le f_\beta\) for \(\alpha\le\beta\) and for each \(f\in C(X)\) there exists \(\alpha\in\omega^\omega\) with \(|f|\le f_\alpha\). In the main part of the paper under review the authors prove that for any Tychonoff space \(X\) the following conditions are equivalent: (1) \(C_p(X)\) is \(K\)-analytic-framed in \(\mathbb{R}^X\); (2) \(X\) admits a nice framing; (3) \(X\) has a fundamental resolution of functions. Some other properties of spaces with nice framing are proved in the next sections of the paper. For example, if \(\mathfrak{M}\) denotes the class of metrizable and separable spaces with a nice framing, then: (a) If \(X\in\mathfrak{M}\), then \(X\) admits a nice framing such that for each \(\alpha\in\omega^\omega\) the layer \(\{ U_{\alpha,n}\colon n\in\omega\}\) consists of compact sets. (b) If \(X\in\mathfrak{M}\) is non-Polish, then \(X\) admits also a nice framing such that for each \(\alpha\in \omega^\omega\) there exists \(n\in\omega\) such that \(U_{\alpha,n}\) is not compact. (c) There exists a countable Polish space \(X\in\mathfrak{M}\) with the conclusion like in item (b).
scientific article; zbMATH DE number 7345659
scientific article