Clopen set


In topology, a clopen set in a topological space is a set which is both open and closed. That this is possible may seem counter-intuitive, as the common meanings of open and closed are antonyms, but their mathematical definitions are not mutually exclusive. A set is closed if its complement is open, which leaves the possibility of an open set whose complement is also open, making both sets both open and closed, and therefore clopen.

Examples

In any topological space X, the empty set and the whole space X are both clopen.
Now consider the space X which consists of the union of the two open intervals and of R. The topology on X is inherited as the subspace topology from the ordinary topology on the real line R. In X, the set is clopen, as is the set. This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen.
Now let X be an infinite set under the discrete metricthat is, two points p, q in X have distance 1 if they're not the same point, and 0 otherwise. Under the resulting metric space, any singleton set is open; hence any set, being the union of single points, is open. Since the complement of any set is therefore closed, all sets in the metric space are clopen.
As a less trivial example, consider the space Q of all rational numbers with their ordinary topology, and the set A of all positive rational numbers whose square is bigger than 2. Using the fact that is not in Q, one can show quite easily that A is a clopen subset of Q.

Properties