On the countable definability of sets

Let $Q$ be a connected set in $\mathbb C^p$ . Denote by $D[Q]$ the set of all domains containing $Q$, and let $W(Q)$ be the set of all convex domains from $D[Q]$. We present tests for classes $D[Q]$ and $W(Q)$ (in the case when $Q$ is convex for the last one) to have a countable basis. The results are expressed in terms of properties of the boundary $\operatorname{Fr}Q$ of the set $Q$.