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

Denote by ${\mathbf{C}}_p[{\mathfrak{M}}_0]$ the $C_p$-stable closure of the class ${\mathfrak{M}}_0$ of all separable metrizable spaces, i.e., $\mathbf C_p[\mathfrak M_0]$isthesmallestclassoftopologicalspacesthatcontains$\mathfrak M_0$ 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 (2014), we prove that ${\mathbf{C}}_p[{\mathfrak{M}}_0]$ coincides with the class of all Tychonoff spaces of cardinality strictly less than ${\mathrm{\beth }}_{{\omega }_1}$. Being motivated by the theory of Generalized Metric Spaces, we characterize also other natural $C_p$-type stable closures of the class ${\mathfrak{M}}_0$.