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Clopen set

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**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.BOOK, Bartle, Robert G., Robert G. Bartle, Sherbert, Donald R., 1992, 1982, Introduction to Real Analysis, 2nd, John Wiley & Sons, Inc., 348, (regarding the real numbers and the empty set in R)BOOK, Hocking, John G., Young, Gail S., 1961, Topology, Dover Publications, Inc., NY, 56, (regarding topological spaces)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 sqrt 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 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.BOOK, Mendelson, Bert, 1990, 1975, Introduction to Topology, Third, Dover, 0-486-66352-3, 87, Let
*A*be a subset of a topological space. Prove that Bdry (*A*) = âˆ… if and only if*A*is open and closed., (Given as Exercise 7) - 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

{{reflist}}## References

- WEB, Morris, Sidney A., Topology Without Tears,weblinkweblink" title="web.archive.org/web/20130419134743weblink">weblink 19 April 2013,

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