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., \(\mathbf C_p[\mathfrak M_0]\) is the smallest class of topological spaces that contains \(\mathfrak M_0\) 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 \(\beth_{\omega_1}\). Being motivated by the theory of Generalized Metric Spaces, we characterize also other natural \(C_p\)-type stable closures of the class \(\mathfrak M_0\).