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geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is ...
,
topology In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, and related branches of mathematics, a closed set is a
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
whose
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-clas ...
is an
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
. In a
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 po ...
, a closed set can be defined as a set which contains all its
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
s. In a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...
, a closed set is a set which is closed under the
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
operation. This should not be confused with a closed manifold.


Equivalent definitions

By definition, a subset A of a
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 po ...
(X, \tau) is called if its complement X \setminus A is an open subset of (X, \tau); that is, if X \setminus A \in \tau. A set is closed in X if and only if it is equal to its closure in X. Equivalently, a set is closed if and only if it contains all of its
limit point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also contai ...
s. Yet another equivalent definition is that a set is closed if and only if it contains all of its boundary points. Every subset A \subseteq X is always contained in its (topological) closure in X, which is denoted by \operatorname_X A; that is, if A \subseteq X then A \subseteq \operatorname_X A. Moreover, A is a closed subset of X if and only if A = \operatorname_X A. An alternative characterization of closed sets is available via
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s and nets. A subset A of a topological space X is closed in X if and only if every
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
of every net of elements of A also belongs to A. In a
first-countable space In topology, a branch of mathematics, a first-countable space is a topological space satisfying the "first axiom of countability". Specifically, a space X is said to be first-countable if each point has a countable neighbourhood basis (local base) ...
(such as a metric space), it is enough to consider only convergent
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is calle ...
s, instead of all nets. One value of this characterization is that it may be used as a definition in the context of
convergence space In mathematics, a convergence space, also called a generalized convergence, is a set together with a relation called a that satisfies certain properties relating elements of ''X'' with the family of filters on ''X''. Convergence spaces generaliz ...
s, which are more general than topological spaces. Notice that this characterization also depends on the surrounding space X, because whether or not a sequence or net converges in X depends on what points are present in X. A point x in X is said to be a subset A \subseteq X if x \in \operatorname_X A (or equivalently, if x belongs to the closure of A in the
topological subspace 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 ...
A \cup \, meaning x \in \operatorname_ A where A \cup \ is endowed with 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 ...
induced on it by XIn particular, whether or not x is close to A depends only on the subspace A \cup \ and not on the whole surrounding space (e.g. X, or any other space containing A \cup \ as a topological subspace).). Because the closure of A in X is thus the set of all points in X that are close to A, this terminology allows for a plain English description of closed subsets: :a subset is closed if and only if it contains every point that is close to it. In terms of net convergence, a point x \in X is close to a subset A if and only if there exists some net (valued) in A that converges to x. If X is a
topological subspace 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 ...
of some other topological space Y, in which case Y is called a of X, then there exist some point in Y \setminus X that is close to A (although not an element of X), which is how it is possible for a subset A \subseteq X to be closed in X but to be closed in the "larger" surrounding super-space Y. If A \subseteq X and if Y is topological super-space of X then A is always a (potentially proper) subset of \operatorname_Y A, which denotes the closure of A in Y; indeed, even if A is a closed subset of X (which happens if and only if A = \operatorname_X A), it is nevertheless still possible for A to be a proper subset of \operatorname_Y A. However, A is a closed subset of X if and only if A = X \cap \operatorname_Y A for some (or equivalently, for every) topological super-space Y of X. Closed sets can also be used to characterize
continuous functions In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
: a map f : X \to Y is
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
if and only if f\left( \operatorname_X A \right) \subseteq \operatorname_Y (f(A)) for every subset A \subseteq X; this can be reworded in
plain English Plain English (or layman's terms) are groups of words that are to be clear and easy to know. It usually avoids the use of rare words and uncommon euphemisms to explain the subject. Plain English wording is intended to be suitable for almost anyone, ...
as: f is continuous if and only if for every subset A \subseteq X, f maps points that are close to A to points that are close to f(A). Similarly, f is continuous at a fixed given point x \in X if and only if whenever x is close to a subset A \subseteq X, then f(x) is close to f(A).


More about closed sets

The notion of closed set is defined above in terms of
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s, a concept that makes sense for
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 po ...
s, as well as for other spaces that carry topological structures, such as
metric space In mathematics, a metric space is a set together with a notion of '' distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general set ...
s, differentiable manifolds,
uniform space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifo ...
s, and
gauge space In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unif ...
s. Whether a set is closed depends on the space in which it is embedded. However, the
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in British ...
Hausdorff space In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the m ...
s are " absolutely closed", in the sense that, if you embed a compact Hausdorff space D in an arbitrary Hausdorff space X, then D will always be a closed subset of X; the "surrounding space" does not matter here.
Stone–Čech compactification In the mathematical discipline of general topology, Stone–Čech compactification (or Čech–Stone compactification) is a technique for constructing a universal map from a topological space ''X'' to a compact Hausdorff space ''βX''. The Ston ...
, a process that turns a
completely regular In topology and related branches of mathematics, Tychonoff spaces and completely regular spaces are kinds of topological spaces. These conditions are examples of separation axioms. A Tychonoff space refers to any completely regular space that is ...
Hausdorff space into a compact Hausdorff space, may be described as adjoining limits of certain nonconvergent nets to the space. Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff space is closed. Closed sets also give a useful characterization of compactness: a topological space X is compact if and only if every collection of nonempty closed subsets of X with empty intersection admits a finite subcollection with empty intersection. A topological space X is disconnected if there exist disjoint, nonempty, open subsets A and B of X whose union is X. Furthermore, X is
totally disconnected In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
if it has an
open basis In mathematics, a base (or basis) for the Topology (structure), topology of a topological space is a Family of sets, family \mathcal of Open set, open subsets of such that every open set of the topology is equal to the set union, union of some S ...
consisting of closed sets.


Properties

A closed set contains its own
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment * ''Boundaries'' (2016 film), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
. In other words, if you are "outside" a closed set, you may move a small amount in any direction and still stay outside the set. Note that this is also true if the boundary is the empty set, e.g. in the metric space of rational numbers, for the set of numbers of which the square is less than 2. * Any intersection of any family of closed sets is closed (this includes intersections of infinitely many closed sets) * 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 ...
of closed sets is closed. * The empty set is closed. * The whole set is closed. In fact, if given a set X and a collection \mathbb \neq \varnothing of subsets of X such that the elements of \mathbb have the properties listed above, then there exists a unique topology \tau on X such that the closed subsets of (X, \tau) are exactly those sets that belong to \mathbb. The intersection property also allows one to define the closure of a set A in a space X, which is defined as the smallest closed subset of X that is a
superset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of A. Specifically, the closure of X can be constructed as the intersection of all of these closed supersets. Sets that can be constructed as the union of
countably In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
many closed sets are denoted Fσ sets. These sets need not be closed.


Examples

* The closed interval
, b The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s is closed. (See for an explanation of the bracket and parenthesis set notation.) * The
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysis ...
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
/math> is closed in the metric space of real numbers, and the set
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\cap \Q of
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 rat ...
s between 0 and 1 (inclusive) is closed in the space of rational numbers, but
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
\cap \Q is not closed in the real numbers. * Some sets are neither open nor closed, for instance the half-open interval ,_1)_in_the_real_numbers. *_Some_sets_are_both_open_and_closed_and_are_called_clopen_sets. *_The_Line_(geometry)#Ray.html" ;"title="clopen_sets.html" ;"title=", 1) in the real numbers. * Some sets are both open and closed and are called clopen sets">, 1) in the real numbers. * Some sets are both open and closed and are called clopen sets. * The Line (geometry)#Ray">ray [1, +\infty) is closed. * The Cantor set is an unusual closed set in the sense that it consists entirely of boundary points and is nowhere dense. * Singleton points (and thus finite sets) are closed in T1 space, T1 spaces and
Hausdorff spaces In topology and related branches of mathematics, a Hausdorff space ( , ), separated space or T2 space is a topological space where, for any two distinct points, there exist neighbourhoods of each which are disjoint from each other. Of the man ...
. * The set of
integers An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
\Z is an infinite and unbounded closed set in the real numbers. * If f : X \to Y is a function between topological spaces then f is continuous if and only if preimages of closed sets in Y are closed in X.


See also

* * * * * * *


Notes


References

* * * * {{DEFAULTSORT:Closed Set General topology