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 lower limit topology, then $\backslash operatorname\_X\; ((0,\; 1))\; =\; [0,\; 1).$
* 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, since every set is closed (and also open), every set is equal to its closure.
* In any indiscrete space $X,$ 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 set, dense.
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 subspace topology, relative topology induced by the Euclidean space $\backslash mathbb,$ and if $S\; =\; \backslash ,$ then $S$ is Clopen set, 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 $\backslash sqrt2$.

topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a ...

$(X,\; \backslash tau)$, 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 ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, the closure of a subset ''S'' of points in a topological space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a ...

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; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

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 $X.$
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, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a ...

s by replacing "open ball" or "ball" with "Topology glossary#N, neighbourhood".
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 a limit point. 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. 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 Adherent point, points of closure of $S.$
- $\backslash operatorname\; S$ is the set $S$ together with Derived set (mathematics), all of its limit points.
- $\backslash operatorname\; S$ is the intersection of all closed set In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...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 (mathematics), net (valued) in $S$ that converges to $x$ in $(X,\; \backslash tau).$

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, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), ''closeness'' is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a ...

:
* 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 topology, standard (metric) topology:
* If $X$ is the Euclidean space $\backslash mathbb$ of real numbers, then $\backslash operatorname\_X\; ((0,\; 1))\; =\; [0,\; 1].$
* If $X$ is the Euclidean space $\backslash mathbb$ then the closure of the set $\backslash mathbb$ of rational numbers is the whole space $\backslash mathbb.$ We say that $\backslash mathbb$ is dense (topology), dense in $\backslash mathbb.$
* If $X$ is the complex number, complex plane $\backslash mathbb\; =\; \backslash mathbb^2,$ then $\backslash operatorname\_X\; \backslash left(\; \backslash \; \backslash right)\; =\; \backslash .$
* If $S$ is a finite set, finite subset of a Euclidean space $X,$ then $\backslash operatorname\_X\; S\; =\; S.$ (For a general topological space, this property is equivalent to the T1 space, TClosure operator

A on a set $X$ is a map (mathematics), mapping of the power set of $X,$ $\backslash mathcal(X)$, into itself which satisfies the Kuratowski closure axioms. Given aclosed set
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space th ...

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 sets of the topology).
The closure operator $\backslash operatorname\_X$ is Duality (mathematics), dual to the Interior (topology), interior 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 Complement (set theory), complements in $X.$
In general, the closure operator does not commute with intersections. However, in a complete metric space the following result does hold:
Facts about closures

A subset $S$ is Closed set, closed in $X$ if and only if $\backslash operatorname\_X\; S\; =\; S.$ In particular: * The closure of the empty set is the empty set; * The closure of $X$ itself is $X.$ * The closure of anintersection
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 of (but need not be equal to) the intersection of the closures of the sets.
* In a union of Finite set, finitely 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 of the union of the closures.
If $S\; \backslash subseteq\; T\; \backslash subseteq\; X$ and if $T$ is a Topological subspace, subspace of $X$ (meaning that $T$ is endowed with the subspace topology 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 topology), 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 $T$ 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 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 topology induced on it by $X$).
This equality is particularly useful when $X$ is a Manifold (mathematics), manifold and the sets in the open cover $\backslash mathcal$ are domains of coordinate charts.
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 set, locally closed in $X$", meaning that if $\backslash mathcal$ is any open cover 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. The powerset of a set $X$ may be realized as a partial order Category (mathematics), category $P$ in which the objects are subsets and the morphisms are inclusion maps $A\; \backslash to\; B$ whenever $A$ is a subset of $B.$ Furthermore, a topology $T$ on $X$ is a subcategory 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 $(A\; \backslash downarrow\; I).$ 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 $\backslash operatorname(A),$ the Interior (topology), interior 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), since all are examples of universal arrows.See also

* * * * *Notes

References

Bibliography

* * * * * * * *External links

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