Some remarks on totally imperfect sets

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 $X \subseteq2^\omega$, if

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 $(\ddagger)$.