I think, it us true in every topological space: if you remove  a set $C$ from a connected component $P$ so that it becomes disconnected: $P\setminus C=\sqcup A_i$, where $A_i$ are connected components, then the connected components of $X\setminus C$ are $A_i$'s and the connected components $Q_j$'s of $X$ which are different from $P$. Thd reason is that the connected components are the inclusion-maximal connected subsets, and each connected subset of $X\setminus C$ lies inside a single connected component of $X$.