ON d-SEPARABILITY OF POWERS AND Cp(X)

A space is called d-separable if it has a dense subset representable as the union of countably many discrete subsets. We answer several problems raised by V. V. Tkachuk by showing that (1) X is d-separable for every T1 space X; (2) if X is compact Hausdorff then X is d-separable; (3) there is a 0-dimensional T2 space X such that X2 is dseparable but X1 (and hence X) is not; (4) there is a 0-dimensional T2 space X such that Cp(X) is not d-separable. The proof of (2) uses the following new result: If X is compact Hausdorff then its square X has a discrete subspace of cardinality d(X). A space is called d-separable if it has a dense subset representable as the union of countably many discrete subsets. Thus d-separable spaces form a common generalization of separable and metrizable spaces. A. V. Arhangelskii was the first to study d-separable spaces in [1], where he proved for instance that any product of d-separable spaces is again d-separable. In [9], V. V. Tkachuk considered conditions under which a function space of the form Cp(X) is d-separable and also raised a number of problems concerning the d-separability of both finite and infinite powers of certain spaces. He again raised some of these problems in his lecture presented at the 2006 Prague Topology Conference. In this note we give solutions to basically all his problems concerning infinite powers and to one concerning Cp(X). Let us start by fixing some notation. As usual, see e. g. [3], we denote the density of a space X by d(X). Also following [3] we use ŝ(X) to denote the smallest cardinal λ such that X has no discrete subspace of size λ. Thus ŝ(X) > κ means that X does have a discrete subspace of cardinality κ. With this we are now in a position to present our first result. 2000 Mathematics Subject Classification. 54A25, 54B10.