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The \(C_p\)-stable closure of the class of separable metrizable spaces. (English) Zbl 1373.54025

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 \(\beth_{\omega_{1}}\).

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
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