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 $E=\bigcap\{E_C: C\in{\cal C}\}$?