On universal sets for small ideals and $\sigma $-ideals

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.