On universal sets for small ideals and $\sigma $-ideals
Bulletin Polish Acad. Sci. Math.
MSC: Primary 03E15; Secondary 03E75
DOI: 10.4064/ba241223-9-9
Published online: 6 October 2025
Abstract
We prove that there exists a closed universal set $E \subseteq X \times Y$ for the nowhere dense sets iff the Polish space $X$ is not $\sigma $-compact. We construct closed universal sets for the collection of finite sets of size $\leq n$ and of size precisely $n$. We develop a technique of construction of universal sets similar to the one of Cieślak and Michalski (2018) but using the properties of the Vietoris topology.
