Embeddings of κ-metrizable spaces into function spaces

Abstract

This paper is motivated by V.V. Uspenskii's results on embeddings of spaces into function spaces and the author's results on countable κ-metrizable spaces. For a Tychonoff topological space Y we denote by C p( 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 C p( 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 G δ-point in X, then X can be embedded into C p( 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 2 ω . As an application, we negatively answer a question posed by A.V. Arkhangel'skii.