COUNTABLE NETWORK WEIGHT AND MULTIPLICATION OF BOREL SETS

A space X Borel multiplies with a space Y if each Borel set of X x Y i s a member of the sigma-algebra in X x Y generated by Borel rectangles . We show that a regular space X Borel multiplies with every regular s pace if and only if X has a countable network. We give an example of a Hausdorff space with a countable network which fails to Borel multipl y with any non-separable metric space. In passing, we obtain a charact erization of those spaces which Borel multiply with the space of count able ordinals, and an internal necessary and sufficient condition for X to Borel multiply with every metric space.