Suppose that $X$ is a connected topological space with a connected subset $S$, and let $K$ denote a component of $X\setminus S$. (All sets listed are assumed non-empty, apart from $\emptyset$ of course.)
Question. Is there an example of sets, as above, such that $\overline S\cap\overline K=\emptyset$?
Remark. If there is such an example, then there is one in which both $S$ and $K$ are closed. Indeed, given any example with $\overline S\cap\overline K=\emptyset$, we may replace $S$ with $\overline S$, and $K$ with $\overline K$, to produce an example with closed $S$ and $K$. Thus we could rephrase our question as follows.
Question. Is there a connected topological space $X$ with a connected and closed subset $S$, and a component $K$ of $X\setminus S$ such that $K$ is closed in $X$, and $S\cap K=\emptyset$ ?
Remark. No such example exists if $X$ is locally-connected. Indeed, in this case $K$ would be both closed in $X$ and (as a component) open, in the open and locally-connected $X\setminus S$, and hence also open in $X$, contradicting that $X$ is connected. (But perhaps the condition that $X$ is locally-connected is not essential. One may assume nice separation axioms.)
Remark. If there is such an example, then $K$ cannot be both closed and open in $X$. Let $p$ be any boundary point of $K$. (Then $p$ belongs to $K$ since $p\not\in S$ and $K\cup\{p\}$ is connected.) Then any neighborhood $U$ of $p$ (and we may assume $U$ disjoint form $S$) would contain a point $q\in U\setminus K$ ( and $q\not\in S$). If $C$ is the component of $q$ in $X\setminus S$ then clearly $\overline C\cap K=\emptyset$ (though perhaps $\overline C\cap S\not=\emptyset$ for some such $C$). We have that $p\in\overline{\bigcup\{C: C \mathrm{\ is\ a\ component\ of\ } X\setminus S \mathrm{\ and\ } C\not= K\}}$ . In particular, $X\setminus S$ must have infinitely many components, if there is an example as in the above question.
I am sure the answer is known, and perhaps it is also easy, I do not know, but I can't think of one quickly. (The books Topology by John G. Hocking and Gail S. Young, and Counterexamples in Topology by Lynn Steen and J. Arthur Seebach, Jr., are two likely books to have a relevant example or a theorem.) The requirement that $X$ is connected is essential, and I need to think if the requirement, that $S$ is connected, is essential. If $S$ it is not connected, perhaps we could "make" it connected by adjoining its cone.
This question is motivated by the following one Components and connectedness by user @samiam , and by my answer to it .
Thank you!
