Uniquely universal sets

We say that X × Y satisfies the Uniquely Universal property (UU) iff there exists an open set U ⊆ X × Y such that for every open set W ⊆ Y there is a unique cross section of U with U x = W . Michael Hrušák raised the question of when does X × Y satisfy UU and noted that if Y is compact, then X must have an isolated point. We consider the problem when the parameter space X is either the Cantor space 2 ω or the Baire space ω ω . We prove the following:1. s a locally compact zero-dimensional Polish space which is not compact, then 2 ω × Y has UU. s Polish, then ω ω × Y has UU iff Y is not compact. s a σ-compact subset of a Polish space which is not compact, then ω ω × Y has UU.