The C_p-stable closure of the class of separable metrizable spaces

Denote by C_p[M₀] the C_p-stable closure of the class M₀ of all separable metrizable spaces, i.e., C_p[M₀] is the smallest class of topological spaces that contains M₀ and is closed under taking subspaces, homeomorphic images, countable topological sums, countable Tychonoff products, and function spaces C_p(X, Y). Using a recent deep result of Chernikov and Shelah, we prove that C_p[M₀] coincides with the class of all Tychonoff spaces of cardinality strictly less than ℶ_ω1. Being motivated by the theory of generalized metric spaces, we also characterize other natural C_p-type stable closures of M₀.