clopen

Clopen is a term in topology describing a subset that is both open and closed in a given topological space. Formally, a set A ⊆ X is clopen if A is open and A is closed. Since the complement of a closed set is open and vice versa, the complement of a clopen set is also clopen. The collection of clopen sets in a space forms a Boolean algebra under union, intersection, and complementation.

Examples illustrate the concept. In the discrete topology on any set, every subset is clopen. In the indiscrete (or trivial) topology, only the empty set and the whole space are clopen. In the real numbers with the usual topology, the only clopen sets are the empty set and the entire real line, because the real line is connected. More generally, a space that is connected has no nontrivial clopen subsets.

Clopen sets and connectedness are closely related: a nonempty clopen subset A of X yields a separation of X into two nonempty open sets, A and X\A. Conversely, if a space can be separated into two nonempty disjoint open sets, each is clopen.

Related concepts include that clopen sets form a Boolean algebra, and that spaces with many clopen sets often have a “zero-dimensional” character. In particular, zero-dimensional spaces have a base consisting of clopen sets, and Stone spaces (compact, zero-dimensional, Hausdorff) have clopen sets providing a natural basis.

A useful quick property: if a set is clopen, its boundary is empty, since its interior equals the set and its closure equals the set.