Bounded resolutions for spaces \(C_p(X)\) and a characterization in terms of \(X\)
type: Item
instance of: scholarly article
title: Bounded resolutions for spaces \(C_p(X)\) and a characterization in terms of \(X\)
zbMATH Open document ID: 1467.54003
published in: Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A: Matemáticas. RACSAM
publication date: 2021-05-07T00:00:00Z
review text: 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 fram…
Mathematics Subject Classification ID: 54C30
Mathematics Subject Classification ID: 54C35
Mathematics Subject Classification ID: 46A03
zbMATH DE Number: 7345659
zbMATH Keywords: cosmic space
zbMATH Keywords: nice framing
zbMATH Keywords: \(K\)-analytic-framed space
zbMATH Keywords: strong framing
author: Juan Carlos Ferrando
author: Jerzy Kąkol
author: W. Śliwa
reviewed by: Tomasz Natkaniec
MaRDI profile type: Publication
full work available at URL: s13398-021-01029-z
OpenAlex ID: W3141862324
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DOI: 10.1007/S13398-021-01029-Z
description: scientific article; zbMATH DE number 7345659