Function spaces and d-separability

The object of this paper is to study when a function space is d-separable, i.e., has a dense &#963-discrete subspace. Several sufficient conditions are obtained for Cp(X) to be d-separable; as an application it is proved that Cp(X) is d-separable for any Corson compact space X. We give a characterization for Cp(X) &#215 Cp(X) to be d-separable and construct, under CH, an example of a non-d-separable space X such that X &#215 X is d-separable. We also establish that if X is a Gul&#146fko space (i.e., Cp(X) is Lindel&#246Nof &#8721) then any subspace of X is d-separable. Keywords: Lindel&#246f &#8721-space, Gul'ko space, d-separable space, condensation, i-weightQuaestiones Mathematicae 28(2005), 409–424

Lindelof Sigma-property in C-p(X) and p(Cp(X)) = omega do not imply countable network weight in X

We prove that there are Tychonoff spaces X for which p(C-p(X)) = w and C-p(X) is a Lindelof C-space while the network weight of X is uncountable. This answers Problem 75 from [4]. An example of a space Y is given such that p(Y) = w and C-p(Y) is a Lindelof Sigma -space, while the network weight of Y is uncountable. This gives a negative answer to Problem 73 from [4]. For a space X with one non-isolated point a necessary and sufficient condition in terms of the topology on X is given for C-p(X) to have countable point-finite cellularity