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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 $$\mathfrak M_0$$.

### Author and article information

###### Journal
06 December 2014
2014-12-09
###### Article
1412.2240