TheInfoList

In
geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mat ... ,
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ... , and related branches of
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a closed set is a set whose
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...
is an
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. In a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
, a closed set can be defined as a set which contains all its
limit point In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...
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 or, alternatively, if every Cauchy sequence in converges in . Intuitively, a space is complet ...
, a closed set is a set which is closed under the
limit Limit or Limits may refer to: Arts and media * Limit (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
operation. This should not be confused with a
closed manifold In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
.

# Equivalent definitions of a closed set

By definition, a subset $A$ of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
$\left(X, \tau\right)$ 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 point In mathematics, a limit point (or cluster point or accumulation point) of a set S in a topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is def ...
s. Yet another equivalent definition is that a set is closed if and only if it contains all of its
boundary points In topology and mathematics in general, the boundary of a subset ''S'' of a topological space ''X'' is the set of points which can be approached both from ''S'' and from the outside of ''S''. More precisely, it is the set of points in the closure ...
. 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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... 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 (music), a way to characterize harmony * Limit (song), "Limit" (song), a 2016 single by Luna Sea * Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...
of every net of elements of $A$ also belongs to $A.$ In a
first-countable space In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
(such as a metric space), it is enough to consider only convergent
sequence In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ... 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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
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 subspaceIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structu ...
$A \cup \,$ meaning $x \in \operatorname_ A$ where $A \cup \$ is endowed with the
subspace topologyIn topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a Topological_space#Definitions, topology induced from that of ''X'' called the subspace topology (or the relative ... 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 subspaceIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structu ...
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 Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
: 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 ga ...
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 Plain English (or layman's terms) is language that is considered to be clear and concise. It may often attempt to avoid the use of uncommon vocabulary and lesser-known euphemisms in order to explain the subject matter. The wording is intended to be ...
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 set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s, a concept that makes sense for
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s, as well as for other spaces that carry topological structures, such as
metric space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gene ...
s,
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...
s,
uniform space In the mathematical field of topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathemati ...
s, and
gauge space In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population i ...
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 A pact, from Latin ''pactum'' ("something agreed upon"), is a formal agreement. In international relations International relations (IR), international affairs (IA) or internationa ...
Hausdorff space In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), ... 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 compactificationIn the mathematical discipline of general topology , a useful example in point-set topology. It is connected but not path-connected. In mathematics, general topology is the branch of topology that deals with the basic Set theory, set-theoretic defin ...
, 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. Tychonoff spaces are named after Andrey Nikolayevich Tychonoff, w ...
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 disconnectedIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
if it has an open basis consisting of closed sets.

# Properties of closed sets

A closed set contains its own
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 ...
. 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 The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
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 #REDIRECT Empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry ... 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 In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 countable set is a Set (mathematics), set with the same cardinality (cardinal number, number of elements) as some subset of the set of natural numbers. A countable set is either a finite set or a countably infinite set. Whether f ...
many closed sets are denoted Fσ sets. These sets need not be closed.

# Examples of closed sets

* The closed interval 
, b The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
/math> of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s is closed. (See for an explanation of the bracket and parenthesis set notation.) * The
unit interval In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
/math> is closed in the metric space of real numbers, and the set 
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
\cap \Q of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s between $0$ and $1$ (inclusive) is closed in the space of rational numbers, but 
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
\cap \Q 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: Science and mathematics * Ray (geometry), half of a line proceeding from an initial point * Ray (graph theory), an infinite sequence of vertices such that each vertex appears at most once in the sequence and each two consecutive ... $\left[1, +\infty\right)$ 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 Hausdorff spaces. * The set of [ ntegers $\Z$ 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 a continuous if and only if preimages of closed sets in $Y$ are closed in $X.$