Uniquely Universal Sets
Logic
2011-06-09 v1
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 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.
Keywords
Cite
@article{arxiv.1106.1629,
title = {Uniquely Universal Sets},
author = {Arnold W. Miller},
journal= {arXiv preprint arXiv:1106.1629},
year = {2011}
}
Comments
LaTex2e: 17 pages Latest version at: http://www.math.wisc.edu/~miller