Embeddings of κ-metrizable spaces into function spaces

This paper is motivated by V.V. Uspenskiiʹs results on embeddings of spaces into function spaces and the authorʹs results on countable κ-metrizable spaces. For a Tychonoff topological space Y we denote by Cp(Y) the space of all real-valued continuous functions on Y with the topology of pointwise convergence. In this paper, we are interested in an “intrinsic” characterization of spaces which can be embedded into Cp(Y) on some compact space Y, and an estimation of the number of countable stratifiable κ-metrizable spaces. We prove that (1) if X is a space with a unique nonisolated point, and the nonisolated point is a Gδ-point in X, then X can be embedded into Cp(Y) for some compact space Y iff X is κ-metrizable in the sense of E.V. Ščepin, (2) the number of countable stratifiable κ-metrizable spaces is 2ω. As an application, we negatively answer a question posed by A.V. Arkhangelʹskii.