  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.
