The $C_p$-stable closure of the class of separable metrizable spaces

TL;DR

This paper characterizes the $C_p$-stable closure of all separable metrizable spaces, showing it equals the class of Tychonoff spaces with cardinality less than $eth_{ ext{omega}_1}$, using advanced set-theoretic results.

Contribution

It precisely identifies the $C_p$-stable closure of separable metrizable spaces as all Tychonoff spaces below a certain cardinality, extending understanding of function space closures.

Findings

The $C_p$-stable closure equals all Tychonoff spaces with cardinality less than $eth_{ ext{omega}_1}$.

Utilizes a deep set-theoretic result by Chernikov and Shelah (2014).

Provides characterizations of other natural $C_p$-type stable closures.

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 $\mathfrak M_0$.