Clopen set Totally Explained

Everything about Clopen Set totally explained

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]. 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 √2 isn't 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's neither open nor closed in R.)
   1. Bartle, Robert G. and Sherbert, Donald R.: Introduction to Real Analysis, 2cd ed. John Wiley & Sons, Inc.1982, 1992, pg 348 regarding the real numbers and the empty set in R
   2. Hocking, John G., Young, Gail S.: Topology, Dover Publications, Inc, NY, 1961 pg 5 and 6 regarding topological spaces

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's 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

"Topology Without Tears" by Sidney A. Morris

Further Information

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