Munkres problem 24.11: If $A$ is a connected subspace of $X$, does it follow that $\operatorname{Int}A$ and $\operatorname{Bd}A$ are connected? Does the converse hold?
I've answered these questions, but I'm wondering if the converse might hold if $X$ satisfies some reasonable conditions, like being normal and connected, or perhaps locally connected, and assuming that neither the interior nor the boundary is empty.