Everywhere $\mathcal{I}$ sets

Abstract

Let $\mathcal{I}$ be a $σ$-ideal on the Cantor space $2^ω$. A set $X\subseteq 2^ω$ is called everywhere $\mathcal{I}$ if for any $S\in [ω]^ω$ the set $X\!\restriction\! S\in\mathcal{I}(2^S).$ We will discuss relations between such families in the class of perfect and Borel sets.