# Clopen set

In topology, a **clopen set** (a portmanteau of **closed-open 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.[1][2]

Now consider the space *X* which consists of the union of the two open 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.

Now let *X* be an infinite set under the discrete metric – that 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**. (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.[3]
- 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.

## Notes

- Bartle, Robert G.; Sherbert, Donald R. (1992) [1982].
*Introduction to Real Analysis*(2nd ed.). John Wiley & Sons, Inc. p. 348. (regarding the real numbers and the empty set in R) - Hocking, John G.; Young, Gail S. (1961).
*Topology*. NY: Dover Publications, Inc. p. 56. (regarding topological spaces) - Mendelson, Bert (1990) [1975].
*Introduction to Topology*(Third ed.). Dover. p. 87. ISBN 0-486-66352-3.Let

(Given as Exercise 7)*A*be a subset of a topological space. Prove that Bdry (*A*) = ∅ if and only if*A*is open and closed.

## References

- Morris, Sidney A. "Topology Without Tears". Archived from the original on 19 April 2013.