Sorry for the "crazy" title, but that's what it is...
Let $X$ be a topological space.
Let $E$ be a subset of $X$.
Let $F$ be a connected component of $E$.
Is the following true? Why?
For every connected component $C$ of $X \setminus F$, there exists a connected component $D$ of $X \setminus E$ such that $D \subseteq C$.
In case there is a counterexample, can we restrict the hypotheses on $X$ and $E$ to make it work? (For example, compactness of $E$ or $X = \mathbb R^2$.)
Thanks in advance for your help.