 If "X" has only finitely many connected components, then each component is the complement of a finite union of closed sets and therefore open.In general, the connected components need not be open, since, e.g., there exist totally disconnected spaces (i.e., formula_24 for all points "x") that are not discrete, like Cantor space.However, the connected components of a locally connected space are also open, and thus are clopen sets.