Application of selection principles in the study of the properties of function spaces

Abstract

For a Tychonoff space X, we denote by Cp(X) the space of all real-valued continuous functions on X with the topology of pointwise convergence. In this paper we prove that: If every finite power of X is Lindelöf then Cp(X) is strongly sequentially separable iff X is γ-set.Bα(X) (= functions of Baire class α (1 < α≤ ω1) on a Tychonoff space X with the pointwise topology) is sequentially separable iff there exists a Baire isomorphism class α from a space X onto a σ-set.Bα(X) is strongly sequentially separable iff iw(X) = ℵ0 and X is a Zα-cover γ-set for 0 < α≤ ω1.There is a consistent example of a set of reals X such that Cp(X) is strongly sequentially separable but B1(X) is not strongly sequentially separable.B(X) is sequentially separable but is not strongly sequentially separable for a b-Sierpiński set X. © 2018, Akadémiai Kiadó, Budapest, Hungary.