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

17 pages

References in corpus (1)

• Networks for the weak topology of Banach and Fréchet spaces

Cited by in corpus (1)

• Lusin and Suslin properties of function spaces