Title: Uniquely Universal Sets
Author: Arnold W. Miller
Abstract:
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 0-dim 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.
LaTex2e: 17 pages
Latest version at: www.math.wisc.edu/~miller