On the $C_k$-stable closure of the class of (separable) metrizable spaces

TL;DR

This paper characterizes the $C_k$-stable closure of metrizable spaces, showing it coincides with spaces embeddable into certain function spaces, and explores the relationships among Ascoli, $eth_0$-spaces, and $rak P$-spaces.

Contribution

It provides a precise description of the $C_k$-stable closure of metrizable spaces and clarifies the embedding properties of Ascoli and $eth_0$-spaces within this framework.

Findings

The class $ extbf{C}_k[ extbf{M}]$ equals spaces homeomorphic to subspaces of $C_k(X,Y)$ with separable metrizable $X$ and metrizable $Y$.

Every Ascoli $eth_0$-space embeds into some $C_k(X,Y)$ with separable metrizable $X,Y$.

The class $ extbf{C}_k[ extbf{M}]$ properly contains all Ascoli $eth_0$-spaces and is contained in $rak P$-spaces.

Abstract

Denote by $\mathbf C_k[\mathfrak M]$ the $C_k$-stable closure of the class $\mathfrak M$ of all metrizable spaces, i.e., $\mathbf C_k[\mathfrak M]$ is the smallest class of topological spaces that contains $\mathfrak M$ and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces $C_k(X,Y)$ with Lindel\"of domain. We show that the class $\mathbf C_k[\mathfrak M]$ coincides with the class of all topological spaces homeomorphic to subspaces of the function spaces $C_k(X,Y)$ with a separable metrizable space $X$ and a metrizable space $Y$. We say that a topological space $Z$ is Ascoli if every compact subset of $C_k(Z)$ is evenly continuous; by the Ascoli Theorem, each $k$-space is Ascoli. We prove that the class $\mathbf C_k[\mathfrak M]$ properly contains the class of all Ascoli $\aleph_0$-spaces and is properly contained in the class of $\mathfrak P$-spaces, recently introduced by Gabriyelyan and K\k{a}kol. Consequently, an Ascoli space $Z$ embeds into the function space $C_k(X,Y)$ for suitable separable metrizable spaces $X$ and $Y$ if and only if $Z$ is an $\aleph_0$-space.