The $C_p$-stable closure of the class of separable metrizable spaces
Volume 146 / 2017
Colloquium Mathematicum 146 (2017), 283-294
MSC: Primary 46E10, 54C35; Secondary 54E18, 12J15, 06A05.
DOI: 10.4064/cm6475-3-2016
Published online: 14 November 2016
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, 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 also characterize other natural $C_p$-type stable closures of $\mathfrak M _0$.
