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. A Möbius strip, a surface with only one side and one edge; such shapes are an object of study in topology. ... Look up Portmanteau word in Wiktionary, the free dictionary. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In topology and related branches of mathematics, a closed set is a set whose complement is open. ...

Examples

In any topological space X, the empty set and the whole space X are both clopen. In mathematics and more specifically set theory, the empty set is the unique set which contains no elements. ...

Now consider the space X which consists of the union of the two intervals [0,1] and [2,3]. 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. In mathematics, interval is a concept relating to the sequence and set-membership of one or more numbers. ... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In mathematics, the real line is simply the set of real numbers. ... Connected and disconnected subspaces of R². The space A at top is connected; the shaded space B at bottom is not. ...

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 √2 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 not closed in R.) In mathematics, a rational number (commonly called a fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ... 2 (two) is the natural number following 1 and preceding 3. ...

Facts

• 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.