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 Hrus̆ák 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 or the Baire space ω. We prove the following: 1. If Y is a locally compact zero dimensional Polish space which is not compact, then 2 × Y has UU. 2. If Y is Polish, then ω × 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 ω × Y has UU. For any space Y with a countable basis there exists an open set U ⊆ 2×Y which is universal for open subsets of Y , i.e., W ⊆ Y is open iff there exists x ∈ 2 with Ux = def {y ∈ Y : (x, y) ∈ U} = W. To see this let {Bn : n < ω} be a basis for Y . Define (x, y) ∈ U iff ∃n (x(n) = 1 and y ∈ Bn). More generally if X contains a homeomorphic copy of 2 then X×Y will have a universal open set. In 1995 Michael Hrušák mentioned the following problem to us. Most of the results in this note were proved in June and July of 2001. MSC2010: 03E15