More on dominated and microscopic sets

Let $S$ be a family of sequences decreasing to zero. A set $E$ in a metric space is $S$-dominated if, for every $s\in S$, there exists a countable cover $\{E_n\}$ of $E$ such that $diam E_n<s_n$ for every $n$. We continue the study of families of dominated sets initiated in the previous paper. We lo