Some remarks on totally imperfect sets

Abstract:

We prove the following two theorems.

Theorem 1. Let $X$ be a strongly meager subset of $2^{\omega \times \omega }$. Then it is dual Ramsey null.

We will say that a $\sigma$-ideal $\mathcal {I}$ of subsets of $2^{\omega }$ satisfies the condition $(\ddagger )$ iff for every $X \subseteq 2^\omega$, if \[ \forall _{f \in \omega ^{\uparrow \omega }} \lbrace g \in \omega ^{\uparrow \omega }\colon \neg (f \prec g) \rbrace \cap X \in \mathcal {I}, \] then $X \in \mathcal {I}$.

Theorem 2. The $\sigma$-ideals of perfectly meager sets, universally meager sets and perfectly meager sets in the transitive sense satisfy the condition $(\ddagger )$.