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      The \(C_p\)-stable closure of the class of separable metrizable spaces

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          Abstract

          Denote by \(\mathbf C_p[\mathfrak M_0]\) the \(C_p\)-stable closure of the class \(\mathfrak M_0\) of all separable metrizable spaces, i.e., \(\mathbf C_p[\mathfrak M_0]\) is the smallest class of topological spaces that contains \(\mathfrak M_0\) and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces \(C_p(X,Y)\). Using a recent deep result of Chernikov and Shelah (2014), we prove that \(\mathbf C_p[\mathfrak M_0]\) coincides with the class of all Tychonoff spaces of cardinality strictly less than \(\beth_{\omega_1}\). Being motivated by the theory of Generalized Metric Spaces, we characterize also other natural \(C_p\)-type stable closures of the class \(\mathfrak M_0\).

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          Author and article information

          Journal
          06 December 2014
          2014-12-09
          Article
          1412.2240

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          Custom metadata
          46E10, 54C35, 54E18, 12J15, 06A05
          7 pages
          math.GN

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