Function spaces and d-separability

Abstract

The object of this paper is to study when a function space is d-separable, i.e., has a dense σ-discrete subspace. Several sufficient conditions are obtained for C _p(X) to be d-separable; as an application it is proved that C _p(X) is d-separable for any Corson compact space X. We give a characterization for C _p(X) × C _p(X) to be d-separable and construct, under CH, an example of a non-d-separable space X such that X × X is d-separable. We also establish that if X is a Gul'ko space (i.e., C _p(X) is Lindelöf Σ) then any subspace of X is d-separable.