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In
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 c ...
,
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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 is an open set. 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 poin ...
, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a closed set is a set which is
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, ...
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 poin ...
$\left(X, \tau\right)$ is called if its complement $X \setminus A$ is an open subset of $\left(X, \tau\right)$; 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 points. 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 called ...
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 (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 called ...
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 $A \cup \,$ meaning $x \in \operatorname_ A$ where $A \cup \$ is endowed with the subspace topology induced on it by $X$In 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 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: 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 g ...
if and only if $f\left\left( \operatorname_X A \right\right) \subseteq \operatorname_Y \left(f\left(A\right)\right)$ for every subset $A \subseteq X$; this can be reworded in plain English 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 $f\left(A\right).$ 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 $f\left(x\right)$ is close to $f\left(A\right).$

The notion of closed set is defined above in terms of open sets, 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 poin ...
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 setti ...
s, differentiable manifolds, uniform spaces, and gauge spaces. 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 Britis ...
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 ma ...
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, a process that turns a completely regular 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 if it has an open basis consisting of closed sets.

# Properties

A closed set contains its own boundary. 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 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 $\left(X, \tau\right)$ 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 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 many closed sets are denoted Fσ sets. These sets need not be closed.

# Examples

* The closed interval
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 ...
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 ra ...
s between $0$ and $1$ (inclusive) is closed in the space of rational numbers, but is not closed in the real numbers. * Some sets are neither open nor closed, for instance the half-open interval _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 Ray may refer to: Fish * Ray (fish), any cartilaginous fish of the superorder Batoidea * Ray (fish fin anatomy), a bony or horny spine on a fin Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (gr ...
$\left[1, +\infty\right)$ 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. * The set of integers $\Z$ 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 $X.$