If $A$ is connected, is at least one of the sets $\mathrm{Int}A$ and $\mathrm{Bd}A$ connected?

Question

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.

Answer 1

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.

Answer 2

Let $D_1, D_2$ be two disjoint closed disks in plane with centers $c_1, c_2$. Let $C \subseteq \mathbb{R}^2 \backslash (D_2 \cup D_2)$ be a countable dense subset. Let $D'_i$ be $D_i$ minus $c_i$. Let $X$ be $C \cup D'_1 \cup D'_2$. Then, interior of $X$ is $D'_1 \cup D'_2$ which is disconnected. Also boundary of $X$ has two isolated points $c_1, c_2$.