If $A$ is connected, is at least one of the sets $\mathrm{Int}A$ and $\mathrm{Bd}A$ connected? If $A$ is a connected subset of $X$, does it follow at least one of the sets $\mathrm{Int}A$ and $\mathrm{Bd}A$ are connected?
I have found counterexamples showing that they not both need to be connected, and was wondering whether this result can be strengthened or not.

Thanks to Daniel Fischer for his correction!
Consider the union $X$ of two full triangles $T,T'$ in the plane that meet at a vertex, and remove a small disk from the interior of $T$. Then $X$ is connected, and neither the interior nor the boundary of $X$ are connected.
