On d-separability of powers and

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 X ) is d-separable for every T 1 space X; s compact Hausdorff then X ω is d-separable; is a 0-dimensional T 2 space X such that X ω 2 is d-separable but X ω 1 (and hence X ω ) is not; is a 0-dimensional T 2 space X such that C p ( X ) is not d-separable. roof of (2) uses the following new result: If X is compact Hausdorff then its square X 2 has a discrete subspace of cardinality d ( X ) .