Clopen

A set may be both open and closed (a clopen set ).

A subset of X may be open, closed, both ( clopen set ), or neither.

Equivalently, a space is connected if the only clopen sets are the whole space and the empty set.

Hence, in any topology, the entire space is simultaneously open and closed (" clopen ").

Let be the connected component of x in a topological space X, and be the intersection of all clopen sets containing x (called quasi-component of x.) Then where the equality holds if X is compact Hausdorff or locally connected.