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. 
Arnold W. Miller
math.LO