In

_{σ} sets. These sets need not be closed.

$[1,\; +\backslash infty)$ 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 Hausdorff spaces.
* The set of [ ntegers $\backslash Z$ is an infinite and unbounded closed set in the real numbers.
* If $f\; :\; X\; \backslash to\; Y$ is a function between topological spaces then $f$ is a continuous if and only if preimages of closed sets in $Y$ are closed in $X.$
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 atopological 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 ...

$(X,\; \backslash tau)$ is called if its complement $X\; \backslash setminus\; A$ is an open subset of $(X,\; \backslash tau)$; that is, if $X\; \backslash setminus\; A\; \backslash in\; \backslash tau.$ 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 $A\; \backslash subseteq\; X$ is always contained in its (topological) closure in $X,$ which is denoted by $\backslash operatorname\_X\; A;$ that is, if $A\; \backslash subseteq\; X$ then $A\; \backslash subseteq\; \backslash operatorname\_X\; A.$ Moreover, $A$ is a closed subset of $X$ if and only if $A\; =\; \backslash operatorname\_X\; A.$
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 $A\; \backslash subseteq\; X$ if $x\; \backslash in\; \backslash operatorname\_X\; A$ (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 ...

$A\; \backslash cup\; \backslash ,$ meaning $x\; \backslash in\; \backslash operatorname\_\; A$ where $A\; \backslash cup\; \backslash $ 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\; \backslash cup\; \backslash $ and not on the whole surrounding space (e.g. $X,$ or any other space containing $A\; \backslash cup\; \backslash $ 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 $x\; \backslash in\; X$ 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 $Y\; \backslash setminus\; X$ that is close to $A$ (although not an element of $X$), which is how it is possible for a subset $A\; \backslash subseteq\; X$ to be closed in $X$ but to be closed in the "larger" surrounding super-space $Y.$
If $A\; \backslash subseteq\; X$ and if $Y$ is topological super-space of $X$ then $A$ is always a (potentially proper) subset of $\backslash operatorname\_Y\; A,$ which denotes the closure of $A$ in $Y;$ indeed, even if $A$ is a closed subset of $X$ (which happens if and only if $A\; =\; \backslash operatorname\_X\; A$), it is nevertheless still possible for $A$ to be a proper subset of $\backslash operatorname\_Y\; A.$ However, $A$ is a closed subset of $X$ if and only if $A\; =\; X\; \backslash cap\; \backslash operatorname\_Y\; A$ 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 $f\; :\; X\; \backslash to\; Y$ 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 $f\backslash left(\; \backslash operatorname\_X\; A\; \backslash right)\; \backslash subseteq\; \backslash operatorname\_Y\; (f(A))$ for every subset $A\; \backslash subseteq\; X$; 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 $A\; \backslash subseteq\; X,$ $f$ maps points that are close to $A$ to points that are close to $f(A).$ Similarly, $f$ is continuous at a fixed given point $x\; \backslash in\; X$ if and only if whenever $x$ is close to a subset $A\; \backslash subseteq\; X,$ then $f(x)$ is close to $f(A).$
More about closed sets

The notion of closed set is defined above in terms ofopen 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 ownboundary
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 $\backslash mathbb\; \backslash neq\; \backslash varnothing$ of subsets of $X$ such that the elements of $\backslash mathbb$ have the properties listed above, then there exists a unique topology $\backslash tau$ on $X$ such that the closed subsets of $(X,\; \backslash tau)$ are exactly those sets that belong to $\backslash mathbb.$
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 FExamples 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 $;\; href="/html/ALL/s/,\_1)$_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
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 ...See also

* * * * * *Notes

References

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