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

Denote by C_k[ M] the C_k-stable closure of the class M of all metrizable spaces, i.e., C_k[ M] is the smallest class of topological spaces that contains 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 in this class. We show that the class C_k[ 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 C_k[ M] properly contains the class of all Ascoli ℵ₀-spaces and is properly contained in the class of 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 ℵ₀-space.