Banakh, Taras; Gabriyelyan, Saak The \(C_p\)-stable closure of the class of separable metrizable spaces. (English) Zbl 1373.54025 Colloq. Math. 146, No. 2, 283-294 (2017). Let \(\mathcal{X}\), \(\mathcal{Y}\) and \(\mathcal{Z}\) be classes of topological spaces. The \(C_{p}^{\mathcal{X},\mathcal{Y}}\)-stable closure of \(\mathcal{Z}\), denoted by \(C_{p}^{\mathcal{X},\mathcal{Y}}[\mathcal{Z}]\), is the smallest class of topological spaces that contains \(\mathcal{Z}\), closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products and for any \(X\in \mathcal{X}\cap \mathcal{Z}\) and \(Y\in \mathcal{Y}\cap \mathcal{Z}\) the function space \(C_{p}(X,Y)\) belongs to \(\mathcal{Z}\). In this paper, the authors describe the \(C_{p}^{\mathcal{X},\mathcal{Y}}\)-stable closures of \(\mathcal{M}_{0}\), the class of all separable metrizable spaces for the cases of \(\mathcal{X}\) and \(\mathcal{Y}\) equal to \(\mathcal{M}_{0}\) or \(\mathcal{I}\) the class of all topological spaces. They prove that \(C_{p}[\mathcal{M}_{0}]\), the \(C_{p}\)-stable closure of \(\mathcal{M}_{0}\), coincides with the class of all Tychonoff spaces of cardinality strictly less than \(\beth_{\omega_{1}}\). Reviewer: Abderrahmane Bouchair (Jijel) Cited in 2 Documents MSC: 54C35 Function spaces in general topology 54E18 \(p\)-spaces, \(M\)-spaces, \(\sigma\)-spaces, etc. 12J15 Ordered fields 06A05 Total orders 46E10 Topological linear spaces of continuous, differentiable or analytic functions Keywords:function space; topology of pointwise convergence; countable network; separately continuous function; ordered field PDFBibTeX XMLCite \textit{T. Banakh} and \textit{S. Gabriyelyan}, Colloq. Math. 146, No. 2, 283--294 (2017; Zbl 1373.54025) Full Text: DOI arXiv References: This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.