Let $(X,\tau)$ be a topological space, $S\subseteq X$ such that there is $x^\*\in X\setminus S$.


Let $E$ be the connected component of $X\setminus S$ that contains $x^\*$. Let ${\cal C}$ be the collection of connected components of $S$. For each $C\in {\cal C}$ let $E\_C$ be the connected component of $X\setminus C$ that contains $x^\*$.


Do we have $E=\bigcap\{E\_C: C\in{\cal C}\}$?



Intersection of complements of connected components
Let $X$ be the circle $S^1$ in $\mathbb C$ and let $S$ be the set $\{1,-1\}$. Let $x^\*=i$ then $E$ is the intersection of $X$ with the upper half plane but the intersection of the $E\_C$ also contains point in the lower half plane.


