Uniquely Universal Sets

TL;DR

This paper investigates the conditions under which product spaces satisfy the Uniquely Universal property, providing new characterizations involving Polish, locally compact, and sigma-compact spaces.

Contribution

It establishes new criteria for UU in product spaces involving Cantor, Baire, and sigma-compact subsets of Polish spaces, extending previous knowledge.

Findings

C x Y has UU for noncompact locally compact Polish Y

B x Y has UU iff Y is noncompact Polish

B x Y has UU for noncompact sigma-compact Y

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.