Let $X$ be a compact metric space satisfying the following condition: for any given positive number $\delta>0$, only finitely many components of $X$ have diameter larger than $\delta$.

For a given component $P$ of $X$, let us remove a connected compact subset $C$ of $P$ such that $P\setminus C$ is disconnected. Consider a disjoint partition $P\setminus C=A\cup B$, with $A$ and $B$ at a positive distance. Suppose that $x\in A$ and $y\in B$.

Are then $x$ and $y$ in different quasi-components of $X\setminus C$? or can we find a separation of $X\setminus C=M\cup N$ such that $x\in M$ and $y\in N$?