I have the following problem: If $A$ is a subset of a connected space $X$, and $C$ is a connected component of $X \setminus A$, is it true that $\partial C \subseteq \partial A$?
Note that I am not sure if this statement is true. If $X = \mathbb{R}^n$, then it is not hard to prove:
Obviously $\partial C$ is disjoint from the interior of $A$. Assume that there is some $x \in \partial C$ that does not lie in $\overline{A}$. Then there is some $\varepsilon>0$ so that $B(x,\varepsilon) \subseteq (\overline{A})^c$. Since $B(x,\varepsilon)$ is connected and intersects $C$, it must be contained in $C$ (by the definition of $C$ as a connected component). But then $x$ is an interior point of $C$, a contradiction.
However, I wonder if the statement is still true for any connected topological space $X$ and not just $\mathbb{R}^n$. So far I have found neither a counterexample or a proof.
This question is kind of similar, and the statement discussed there has an elementary topological proof, which gives me a bit if hope for my problem.
