A subset of a topological space is called clopen if it is both open and closed. Equivalently, a clopen set is a complemented element of the frame of open subsets, a definition which makes as good sense for locales as for spaces.
The set of clopen sets in any space forms a Boolean algebra. If the space is a Stone space, then it can be reconstructed from its Boolean algebra of clopens. On the other hand, a space is connected just when the only clopens are the empty set and the whole space.