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

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 Lindelöf 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 Kąkol. 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.

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