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In
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, a clopen set (a
portmanteau A portmanteau word, or portmanteau (, ) is a blend of wordstopological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
is a set which is both
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...
and
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
. That this is possible may seem counter-intuitive, as the common meanings of and 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 closed, and therefore clopen. As described by topologist
James Munkres James Raymond Munkres (born August 18, 1930) is a Professor Emeritus of mathematics at MIT and the author of several texts in the area of topology, including ''Topology'' (an undergraduate-level text), ''Analysis on Manifolds'', ''Elements of Algeb ...
, unlike a
door A door is a hinged or otherwise movable barrier that allows ingress (entry) into and egress (exit) from an enclosure. The created opening in the wall is a ''doorway'' or ''portal''. A door's essential and primary purpose is to provide security b ...
, "a set can be open, or closed, or both, or neither!" emphasizing that the meaning of "open"/"closed" for is unrelated to their meaning for (and so the open/closed door dichotomy does not transfer to open/closed sets). This contrast to doors gave the class of topological spaces known as "
door space In mathematics, in the field of topology, a topological space is said to be a door space if every subset is open or closed (or both Both may refer to: Common English word * ''both'', a determiner or indefinite pronoun denoting two of somethin ...
s" their name.


Examples

In any topological space X, the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
and the whole space X are both clopen. 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 In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
\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 metricthat 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 any set is open, the complement of any set is open too, and therefore any set is closed. So, all sets in this metric space are clopen. As a less trivial example, consider the space \Q of all
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s 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. (A is 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 In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
and X itself. * A set is clopen if and only if its boundary is empty. (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 Discrete may refer to: *Discrete particle or quantum in physics, for example in quantum theory * Discrete device, an electronic component with just one circuit element, either passive or active, other than an integrated circuit *Discrete group, a ...
if and only if all of its subsets are clopen. * Using the
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
and
intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
as operations, the clopen subsets of a given topological space X form a
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in e ...
. Boolean algebra can be obtained in this way from a suitable topological space: see
Stone's representation theorem for Boolean algebras In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first hal ...
.


See also

* *


Notes


References

* * {{cite web, last=Morris, first=Sidney A., title=Topology Without Tears, url=http://uob-community.ballarat.edu.au/~smorris/topology.htm, archive-url=https://web.archive.org/web/20130419134743/http://uob-community.ballarat.edu.au/~smorris/topology.htm, archive-date=19 April 2013 General topology