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)Xd(X) is d-separable for every T1 space X;(2)if X is compact Hausdorff then Xω is d-separable;(3)there is a 0-dimensional T2 space X such that Xω2 is d-separable but Xω1 (and hence Xω) 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 X2 has a discrete subspace of cardinality d(X).