Countable network weight and multiplication of Borel sets

Abstract

A space X X Borel multiplies with a space Y Y if each Borel set of X × Y X\times Y is a member of the σ \sigma -algebra in X × Y X\times Y generated by Borel rectangles. We show that a regular space X X Borel multiplies with every regular space if and only if X X has a countable network. We give an example of a Hausdorff space with a countable network which fails to Borel multiply with any non-separable metric space. In passing, we obtain a characterization of those spaces which Borel multiply with the space of countable ordinals, and an internal necessary and sufficient condition for X X to Borel multiply with every metric space.