Clopen set

In topology, a clopen set (or closed-open set, a portmanteau word) in a topological space is a set which is both open and closed.

Examples

In any topological space X, the empty set and the whole space X are both clopen.^[1][2]

Now consider the space X which consists of the union of the two intervals [0,1] and [2,3] 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 [0,1] is clopen, as is the set [2,3]. 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.

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. (Note also that A is not a clopen subset of the real line R; it is neither open nor closed in R.)

Properties

• A topological space X is connected if and only if the only clopen sets are the empty set and X.
• A set is clopen if and only if its boundary is empty.
• Any clopen set is a union of (possibly infinitely many) connected components.
• If all connected components of X are open (for instance, if X has only finitely many components, or if X is locally connected), then a set is clopen in X if and only if it is a union of connected components.
• A topological space X is discrete if and only if all of its subsets are clopen.
• Using the union and intersection as operations, the clopen subsets of a given topological space X form a Boolean algebra. Every Boolean algebra can be obtained in this way from a suitable topological space: see Stone's representation theorem for Boolean algebras.

Sources

• Sidney A. Morris. "Topology Without Tears". http://uob-community.ballarat.edu.au/~smorris/topology.htm. 

References