Uniquely Universal Sets

We say that X x Y satisfies the Uniquely Universal property (UU) iff there exists a set U open in X x Y such that for every open set W in Y there is a unique cross section U_x of U with U_x=W. Michael Hrusak raised the question of when does X x Y satisfy UU and noted that if Y is compact then X must have an isolated point. We prove the following:

1. If Y is a locally compact noncompact Polish space, then C x Y has UU where C is the Cantor space.
2. If Y is Polish, then B x Y has UU iff Y is not compact where B is the Baire space.
3. If Y is a sigma-compact subset of a Polish space which is not compact, then B x Y has UU.