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 words[topological 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 ...](_blank)
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. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical definitions are not
mutually exclusive
In logic and probability theory, two events (or propositions) are mutually exclusive or disjoint if they cannot both occur at the same time. A clear example is the set of outcomes of a single coin toss, which can result in either heads or tails ...
. A set is closed if its
complement
A complement is something that completes something else.
Complement may refer specifically to:
The arts
* Complement (music), an interval that, when added to another, spans an octave
** Aggregate complementation, the separation of pitch-class ...
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 Alge ...
, 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 spaces" their name.
Examples
In any topological space
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
are both clopen.
Now consider the space
which consists of the union of the two open
intervals
and
of
The topology on
is inherited as the
subspace topology
In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
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 ...
In
the set
is clopen, as is the set
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
be an infinite set under the discrete metricthat is, two points
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
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
of all positive rational numbers whose square is bigger than 2. Using the fact that
is not in
one can show quite easily that
is a clopen subset of
(
is a clopen subset of the real line
; it is neither open nor closed in
)
Properties
* A topological space
is
connected
Connected may refer to:
Film and television
* ''Connected'' (2008 film), a Hong Kong remake of the American movie ''Cellular''
* '' Connected: An Autoblogography About Love, Death & Technology'', a 2011 documentary film
* ''Connected'' (2015 TV ...
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
itself.
* A set is clopen if and only if its
boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
*Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film
*Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
is empty.
[ (Given as Exercise 7)]
* Any clopen set is a union of (possibly infinitely many)
connected components.
* If all
connected components of
are open (for instance, if
has only finitely many components, or if
is
locally connected
In topology and other branches of mathematics, a topological space ''X'' is
locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets.
Background
Throughout the history of topology, connectedness a ...
), then a set is clopen in
if and only if it is a union of connected components.
* A topological space
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 as operations, the clopen subsets of a given topological space
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