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 Ux=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 2ω or the Baire space ωω. We prove the following:1.If Y is a locally compact zero-dimensional Polish space which is not compact, then 2ω×Y has UU.2.If Y is Polish, then ωω×Y has UU iff Y is not compact.3.If Y is a σ-compact subset of a Polish space which is not compact, then ωω×Y has UU.