In

_{1} axiom.)
On the set of real numbers one can put other topologies rather than the standard one.
* If $X\; =\; \backslash mathbb$ is endowed with the

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 ...

$(X,\; \backslash tau)$, the topological closure induces a function $\backslash operatorname\_X\; :\; \backslash wp(X)\; \backslash to\; \backslash wp(X)$ that is defined by sending a subset $S\; \backslash subseteq\; X$ to $\backslash operatorname\_X\; S,$ where the notation $\backslash overline$ or $S^$ may be used instead. Conversely, if $\backslash mathbb$ is a closure operator on a set $X,$ then a topological space is obtained by defining the

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 ...

, the closure of a subset ''S'' of points 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 ...

consists of all point
Point or points may refer to:
Places
* Point, LewisImage:Point Western Isles NASA World Wind.png, Satellite image of Point
Point ( gd, An Rubha), also known as the Eye Peninsula, is a peninsula some 11 km long in the Outer Hebrides (or Western I ...

s in ''S'' together with all limit points
In mathematics, a limit point (or cluster point or accumulation point) of a Set (mathematics), 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 (mathematics), neighbourhood ...

of ''S''. The closure of ''S'' may equivalently be defined as the union of ''S'' and 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 ...

, and also as the 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 all closed set
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 ...

s containing ''S''. Intuitively, the closure can be thought of as all the points that are either in ''S'' or "near" ''S''. A point which is in the closure of ''S'' is a point of closure of ''S''. The notion of closure is in many ways dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
** . . . see more cases in :Duality theories
* Dual ...

to the notion of interior
Interior may refer to:
Arts and media
* Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* The Interior (novel) ...

.
Definitions

Point of closure

For $S$ a subset of aEuclidean space
Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...

, $x$ is a point of closure of $S$ if every open ball
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 ...

centered at $x$ contains a point of $S$ (this point may be $x$ itself).
This definition generalizes to any subset $S$ of a 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 ...

$X.$
Fully expressed, for $X$ a metric space with metric $d,$ $x$ is a point of closure of $S$ if for every $r\; >\; 0$ there exists some $s\; \backslash in\; S$ such that the distance $d(x,\; s)\; <\; r$ (again, $x\; =\; s$ is allowed).
Another way to express this is to say that $x$ is a point of closure of $S$ if the distance $d(x,\; S)\; :=\; \backslash inf\_\; d(x,\; s)\; =\; 0.$
This definition generalizes to 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 by replacing "open ball" or "ball" with "neighbourhood
A neighbourhood (British English
British English (BrE) is the standard dialect of the English language
English is a West Germanic languages, West Germanic language first spoken in History of Anglo-Saxon England, early medieval ...

".
Let $S$ be a subset of a topological space $X.$
Then $x$ is a or of $S$ if every neighbourhood of $x$ contains a point of $S.$
Note that this definition does not depend upon whether neighbourhoods are required to be open.
Limit point

The definition of a point of closure is closely related to the definition of alimit 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 ...

.
The difference between the two definitions is subtle but important – namely, in the definition of limit point, every neighbourhood of the point $x$ in question must contain a point of the set .
The set of all limit points of a set $S$ is called the
Thus, every limit point is a point of closure, but not every point of closure is a limit point.
A point of closure which is not a limit point is an isolated point 400px, "0" is an isolated point of A = ∪ , 2In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus ...

.
In other words, a point $x$ is an isolated point of $S$ if it is an element of $S$ and if there is a neighbourhood of $x$ which contains no other points of $S$ other than $x$ itself.
For a given set $S$ and point $x,$ $x$ is a point of closure of $S$ if and only if $x$ is an element of $S$ or $x$ is a limit point of $S$ (or both).
Closure of a set

The of a subset $S$ of a topological space $(X,\; \backslash tau),$ denoted by $\backslash operatorname\_\; S$ or possibly by $\backslash operatorname\_X\; S$ (if $\backslash tau$ is understood), where if both $X$ and $\backslash tau$ are clear from context then it may also be denoted by $\backslash operatorname\; S,$ $\backslash overline,$ or $S\; ^$ (moreover, $\backslash operatorname$ is sometimes capitalized to $\backslash operatorname$) can be defined using any of the following equivalent definitions:- $\backslash operatorname\; S$ is the set of all points of closure of $S.$
- $\backslash operatorname\; S$ is the set $S$ together with all of its limit points.
- $\backslash operatorname\; S$ is the intersection of all closed set 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 ...s containing $S.$
- $\backslash operatorname\; S$ is the smallest closed set containing $S.$
- $\backslash operatorname\; S$ is the union of $S$ and 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 ...$\backslash partial(S).$
- $\backslash operatorname\; S$ is the set of all $x\; \backslash in\; X$ for which there exists a net Net or net may refer to: Mathematics and physics * Net (mathematics) In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, ...(valued) in $S$ that converges to $x$ in $(X,\; \backslash tau).$

if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...

$S\; =\; \backslash operatorname\; S$
* If $S\; \backslash subseteq\; T$ then $\backslash operatorname\; S$ is a subset of $\backslash operatorname\; T.$
* If $A$ is a closed set, then $A$ contains $S$ if and only if $A$ contains $\backslash operatorname\; S.$
Sometimes the second or third property above is taken as the of the topological closure, which still make sense when applied to other types of closures (see below).
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
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 ...

), $\backslash operatorname\; S$ is the set of all limits
Limit or Limits may refer to:
Arts and media
* Limit (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre (music), genre of music, or the harmonies that can be made using a particular ...

of all 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 of points in $S.$
For a general topological space, this statement remains true if one replaces "sequence" by "net
Net or net may refer to:
Mathematics and physics
* Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence. In essence, ...

" or "filter
Filter, filtering or filters may refer to:
Science and technology Device
* Filter (chemistry), a device which separates solids from fluids (liquids or gases) by adding a medium through which only the fluid can pass
** Filter (aquarium), critical ...

".
Note that these properties are also satisfied if "closure", "superset", "intersection", "contains/containing", "smallest" and "closed" are replaced by "interior", "subset", "union", "contained in", "largest", and "open".
For more on this matter, see closure operatorIn 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 ha ...

below.
Examples

Consider a sphere in 3 dimensions. Implicitly there are two regions of interest created by this sphere; the sphere itself and its interior (which is called an open 3-ball). It is useful to be able to distinguish between the interior of 3-ball and the surface, so we distinguish between the open 3-ball, and the closed 3-ball – the closure of the 3-ball. The closure of the open 3-ball is the open 3-ball plus the surface. Intopological 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 ...

:
* In any space, $\backslash varnothing\; =\; \backslash operatorname\; \backslash varnothing.$
* In any space $X,$ $X\; =\; \backslash operatorname\; X.$
Giving $\backslash mathbb$ and $\backslash mathbb$ the standard (metric) topology:
* If $X$ is the Euclidean space $\backslash mathbb$ 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, then $\backslash operatorname\_X\; ((0,\; 1))\; =$, 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, ...

* If $X$ is the Euclidean space $\backslash mathbb$ then the closure of the set $\backslash mathbb$ 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 is the whole space $\backslash mathbb.$ We say that $\backslash mathbb$ is dense
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass
Mass is both a property
Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...

in $\backslash mathbb.$
* If $X$ is the complex plane
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 ge ...

$\backslash mathbb\; =\; \backslash mathbb^2,$ then $\backslash operatorname\_X\; \backslash left(\; \backslash \; \backslash right)\; =\; \backslash .$
* If $S$ is a finite
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

subset of a Euclidean space $X,$ then $\backslash operatorname\_X\; S\; =\; S.$ (For a general topological space, this property is equivalent to the Tlower limit topology
In mathematics, the lower limit topology or right half-open interval topology is a topological space, topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and h ...

, then $\backslash operatorname\_X\; ((0,\; 1))\; =;\; href="/html/ALL/s/,\_1).$
* If one considers on $X\; =\; \backslash mathbb$ the discrete topology in which every set is closed (open), then $\backslash operatorname\_X\; ((0,\; 1))\; =\; (0,\; 1).$
* If one considers on $X\; =\; \backslash mathbb$ the trivial topology in which the only closed (open) sets are the empty set and $\backslash mathbb$ itself, then $\backslash operatorname\_X\; ((0,\; 1))\; =\; \backslash mathbb.$
These examples show that the closure of a set depends upon the topology of the underlying space. The last two examples are special cases of the following.
* In any discrete space
In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are ''Isolated point, isolated'' from each other in a certain sense. The discrete topology is t ...

, since every set is closed (and also open), every set is equal to its closure.
* In any indiscrete spaceIn topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, or codiscrete. Intuitively, this has the consequence that a ...

$X,$ since the only closed sets are the empty set and $X$ itself, we have that the closure of the empty set is the empty set, and for every non-empty subset $A$ of $X,$ $\backslash operatorname\_X\; A\; =\; X.$ In other words, every non-empty subset of an indiscrete space is dense
The density (more precisely, the volumetric mass density; also known as specific mass), of a substance is its mass
Mass is both a property
Property (''latin: Res Privata'') in the Abstract and concrete, abstract is what belongs to or ...

.
The closure of a set also depends upon in which space we are taking the closure. For example, if $X$ is the set of rational numbers, with the usual relative topologyIn 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 ...

induced by the Euclidean space $\backslash mathbb,$ and if $S\; =\; \backslash ,$ then $S$ is both closed and open in $\backslash mathbb$ because neither $S$ nor its complement can contain $\backslash sqrt2$, which would be the lower bound of $S$, but cannot be in $S$ because $\backslash sqrt2$ is irrational. So, $S$ has no well defined closure due to boundary elements not being in $\backslash mathbb$. However, if we instead define $X$ to be the set of real numbers and define the interval in the same way then the closure of that interval is well defined and would be the set of all greater than $\backslash sqrt2$.
Closure operator

A on a set $X$ is amapping
Mapping may refer to:
* Mapping (cartography), the process of making a map
* Mapping (mathematics), a synonym for a mathematical function and its generalizations
** Mapping (logic), a synonym for functional predicate
Types of mapping
* Animated ...

of the power 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 an ...

of $X,$ $\backslash mathcal(X)$, into itself which satisfies the Kuratowski closure axioms
Kazimierz Kuratowski (; 2 February 1896 – 18 June 1980) was a Polish mathematician
A mathematician is someone who uses an extensive knowledge of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as ...

.
Given a closed set
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 ...

s as being exactly those subsets $S\; \backslash subseteq\; X$ that satisfy $\backslash mathbb(S)\; =\; S$ (so complements in $X$ of these subsets form the 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 of the topology).
The closure operator $\backslash operatorname\_X$ is dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
** . . . see more cases in :Duality theories
* Dual ...

to the interior
Interior may refer to:
Arts and media
* Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* The Interior (novel) ...

operator, which is denoted by $\backslash operatorname\_X,$ in the sense that
:$\backslash operatorname\_X\; S\; =\; X\; \backslash setminus\; \backslash operatorname\_X\; (X\; \backslash setminus\; S),$
and also
:$\backslash operatorname\_X\; S\; =\; X\; \backslash setminus\; \backslash operatorname\_X\; (X\; \backslash setminus\; S).$
Therefore, the abstract theory of closure operators and the Kuratowski closure axioms can be readily translated into the language of interior operators by replacing sets with their complements in $X.$
In general, the closure operator does not commute with intersections. However, in a complete metric space
In mathematical analysis
Analysis is the branch of mathematics dealing with Limit (mathematics), limits
and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...

the following result does hold:
Facts about closures

A subset $S$ is closed in $X$ if and only if $\backslash operatorname\_X\; S\; =\; S.$ In particular: * The closure of theempty 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 the empty set;
* The closure of $X$ itself is $X.$
* The closure of an 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 sets is always a subset
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 (but need not be equal to) the intersection of the closures of the sets.
* In a union of finite
Finite is the opposite of Infinity, infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected ...

ly many sets, the closure of the union and the union of the closures are equal; the union of zero sets is the empty set, and so this statement contains the earlier statement about the closure of the empty set as a special case.
* The closure of the union of infinitely many sets need not equal the union of the closures, but it is always 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 the union of the closures.
If $S\; \backslash subseteq\; T\; \backslash subseteq\; X$ and if $T$ is a subspace of $X$ (meaning that $T$ 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 ...

that $X$ induces on it), then $\backslash operatorname\_T\; S\; \backslash subseteq\; \backslash operatorname\_X\; S$ and the closure of $S$ computed in $T$ is equal to the intersection of $T$ and the closure of $S$ computed in $X$:
:$\backslash operatorname\_T\; S\; ~=~\; T\; \backslash cap\; \backslash operatorname\_X\; S.$Because $\backslash operatorname\_X\; S$ is a closed subset of $X,$ the intersection $T\; \backslash cap\; \backslash operatorname\_X\; S$ is a closed subset of $T$ (by definition of 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 ...

), which implies that $\backslash operatorname\_T\; S\; \backslash subseteq\; T\; \backslash cap\; \backslash operatorname\_X\; S$ (because $\backslash operatorname\_T\; S$ is the closed subset of $T$ containing $S$). Because $T\; \backslash cap\; \backslash operatorname\_X\; S$ is a closed subset of $T,$ from the definition of the subspace topology, there must exist some set $C\; \backslash subseteq\; X$ such that $C$ is closed in $X$ and $\backslash operatorname\_T\; S\; =\; T\; \backslash cap\; C.$ Because $S\; \backslash subseteq\; \backslash operatorname\_T\; S\; \backslash subseteq\; C$ and $C$ is closed in $X,$ the minimality of $\backslash operatorname\_X\; S$ implies that $\backslash operatorname\_X\; S\; \backslash subseteq\; C.$ Intersecting both sides with $T$ shows that $T\; \backslash cap\; \backslash operatorname\_X\; S\; \backslash subseteq\; T\; \backslash cap\; C\; =\; \backslash operatorname\_T\; S.$ $\backslash blacksquare$
In particular, $S$ is dense in $T$ if and only if
In logic
Logic is an interdisciplinary field which studies truth and reasoning
Reason is the capacity of consciously making sense of things, applying logic
Logic (from Ancient Greek, Greek: grc, wikt:λογική, λογική, l ...

$T$ is a subset of $\backslash operatorname\_X\; S.$
If $S,\; T\; \backslash subseteq\; X$ but $S$ is not necessarily a subset of $T$ then only
:$\backslash operatorname\_T\; (S\; \backslash cap\; T)\; ~\backslash subseteq~\; T\; \backslash cap\; \backslash operatorname\_X\; S$
is guaranteed in general, where this containment could be strict (consider for instance $X\; =\; \backslash R$ with the usual topology, $T\; =\; (-\backslash infty,\; 0],$ and $S\; =\; (0,\; \backslash infty)$From $T\; :=\; (-\backslash infty,\; 0]$ and $S\; :=\; (0,\; \backslash infty)$ it follows that $S\; \backslash cap\; T\; =\; \backslash varnothing$ and $\backslash operatorname\_X\; S\; =\; [0,\; \backslash infty),$ which implies
:$\backslash varnothing\; ~=~\; \backslash operatorname\_T\; (S\; \backslash cap\; T)\; ~\backslash neq~\; T\; \backslash cap\; \backslash operatorname\_X\; S\; ~=~\; \backslash .$
) although if $T$ is an open subset of $X$ then the equality $\backslash operatorname\_T\; (S\; \backslash cap\; T)\; =\; T\; \backslash cap\; \backslash operatorname\_X\; S$ will holdLet $S,\; T\; \backslash subseteq\; X$ and assume that $T$ is open in $X.$ Let $C\; :=\; \backslash operatorname\_T\; (T\; \backslash cap\; S),$ which is equal to $T\; \backslash cap\; \backslash operatorname\_X\; (T\; \backslash cap\; S)$ (because $T\; \backslash cap\; S\; \backslash subseteq\; T\; \backslash subseteq\; X$). The complement $T\; \backslash setminus\; C$ is open in $T,$ where $T$ being open in $X$ now implies that $T\; \backslash setminus\; C$ is also open in $X.$ Consequently $X\; \backslash setminus\; (T\; \backslash setminus\; C)\; =\; (X\; \backslash setminus\; T)\; \backslash cup\; C$ is a closed subset of $X$ where $(X\; \backslash setminus\; T)\; \backslash cup\; C$ contains $S$ as a subset (because if $s\; \backslash in\; S$ is in $T$ then $s\; \backslash in\; T\; \backslash cap\; S\; \backslash subseteq\; \backslash operatorname\_T\; (T\; \backslash cap\; S)\; =\; C$), which implies that $\backslash operatorname\_X\; S\; \backslash subseteq\; (X\; \backslash setminus\; T)\; \backslash cup\; C.$ Intersecting both sides with $T$ proves that $T\; \backslash cap\; \backslash operatorname\_X\; S\; \backslash subseteq\; T\; \backslash cap\; C\; =\; C.$ The reverse inclusion follows from $C\; \backslash subseteq\; \backslash operatorname\_X\; (T\; \backslash cap\; S)\; \backslash subseteq\; \backslash operatorname\_X\; S.$ $\backslash blacksquare$ (no matter the relationship between $S$ and $T$).
Consequently, if $\backslash mathcal$ is any open cover
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 ...

of $X$ and if $S\; \backslash subseteq\; X$ is any subset then:
:$\backslash operatorname\_X\; S\; =\; \backslash bigcup\_\; \backslash operatorname\_U\; (U\; \backslash cap\; S)$
because $\backslash operatorname\_U\; (S\; \backslash cap\; U)\; =\; U\; \backslash cap\; \backslash operatorname\_X\; S$ for every $U\; \backslash in\; \backslash mathcal$ (where every $U\; \backslash in\; \backslash mathcal$ 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 $X$).
This equality is particularly useful when $X$ is a 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 surface.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

and the sets in the open cover $\backslash mathcal$ are domains of coordinate chartIn topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real numbers, real ''n''-dimension (mathematics), dimensional Euclidean space. Topological manifolds are an important class of topological spa ...

s.
In words, this result shows that the closure in $X$ of any subset $S\; \backslash subseteq\; X$ can be computed "locally" in the sets of any open cover of $X$ and then unioned together.
In this way, this result can be viewed as the analogue of the well-known fact that a subset $S\; \backslash subseteq\; X$ is closed in $X$ if and only if it is "locally closed
This is a glossary of some terms used in the branch of mathematics known as topology. Although there is no absolute distinction between different areas of topology, the focus here is on general topology. The following definitions are also fundame ...

in $X$", meaning that if $\backslash mathcal$ is any open cover
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 ...

of $X$ then $S$ is closed in $X$ if and only if $S\; \backslash cap\; U$ is closed in $U$ for every $U\; \backslash in\; \backslash mathcal.$
Categorical interpretation

One may elegantly define the closure operator in terms of universal arrows, as follows. Thepowerset
Image:Hasse diagram of powerset of 3.svg, 250px, The elements of the power set of order theory, ordered with respect to Inclusion (set theory), inclusion.
In mathematics, the power set (or powerset) of a Set (mathematics), set is the set of al ...

of a set $X$ may be realized as a partial order
upright=1.15, Fig.1 The set of all subsets of a three-element set \, ordered by set inclusion">inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, affirmative action to change the circumstances and habits that leads to s ...

category
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...

$P$ in which the objects are subsets and the morphisms are inclusion map
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 $A\; \backslash to\; B$ whenever $A$ is a subset of $B.$ Furthermore, a topology $T$ on $X$ is a subcategory
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 ...

of $P$ with inclusion functor $I\; :\; T\; \backslash to\; P.$ The set of closed subsets containing a fixed subset $A\; \backslash subseteq\; X$ can be identified with the comma category
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 ...

$(A\; \backslash downarrow\; I).$ This category — also a partial order — then has initial object $\backslash operatorname\; A.$ Thus there is a universal arrow from $A$ to $I,$ given by the inclusion $A\; \backslash to\; \backslash operatorname\; A.$
Similarly, since every closed set containing $X\; \backslash setminus\; A$ corresponds with an open set contained in $A$ we can interpret the category $(I\; \backslash downarrow\; X\; \backslash setminus\; A)$ as the set of open subsets contained in $A,$ with terminal object
In category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled d ...

$\backslash operatorname(A),$ the interior
Interior may refer to:
Arts and media
* Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas
* Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck
* The Interior (novel) ...

of $A.$
All properties of the closure can be derived from this definition and a few properties of the above categories. Moreover, this definition makes precise the analogy between the topological closure and other types of closures (for example algebraic closure
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 ...

), since all are examples of universal arrows.
See also

* * * * *Notes

References

Bibliography

* * * * * * * *External links

* {{DEFAULTSORT:Closure (Topology) General topology Closure operators