In mathematics, the closure of a subset ''S'' of points in a topological space consists of all points in ''S'' together with all limit points of ''S''. The closure of ''S'' may equivalently be defined as the union of ''S'' and its boundary, and also as the intersection of all closed sets 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 to the notion of interior.

Definitions

** Point of closure **

For $S$ a subset of a Euclidean space, $x$ is a point of closure of $S$ if every open ball 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 spaces by replacing "open ball" or "ball" with "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:

** 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.
In topological space:
* 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 numbers, then $\backslash operatorname\_X\; ((0,\; 1))\; =,\; 1$
* 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 in $\backslash mathbb.$
* If $X$ is the complex plane $\backslash mathbb\; =\; \backslash mathbb^2,$ then $\backslash operatorname\_X\; \backslash left(\; \backslash \; \backslash right)\; =\; \backslash .$
* If $S$ is a finite subset of a Euclidean space $X,$ then $\backslash operatorname\_X\; S\; =\; S.$ (For a general topological space, this property is equivalent to the T_{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 $\backslash operatorname\_X\; ((0,\; 1))\; =$
* 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.
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 topology 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 a mapping of the power set of $X,$ $\backslash mathcal(X)$, into itself which satisfies the Kuratowski closure axioms.
Given a topological space $(X,\; \backslash tau)$, 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 closed sets 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 dual to the 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 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 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 an intersection 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 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 $A$ is a subspace of $X$ containing $S$, then the closure of $S$ computed in $A$ is equal to the intersection of $A$ and the closure of $S$ computed in $X$: $\backslash operatorname\_A\; S\; =\; A\; \backslash cap\; \backslash operatorname\_X\; S.$ In particular, $S$ is dense in $A$ if and only if $A$ is a subset of $\backslash operatorname\_X\; S.$

** 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 $P$ in which the objects are subsets and the morphisms are inclusions $A\; \backslash to\; B$ 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 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), since all are examples of universal arrows.

** See also **

*
*
*
*
*

** Notes **

** References **

*
*
*
*
*
*
*
*

External links

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

Definitions

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 sets containing $S.$
- $\backslash operatorname\; S$ is the smallest closed set containing $S.$
- $\backslash operatorname\; S$ is the union of $S$ and its boundary $\backslash partial(S).$
- $\backslash operatorname\; S$ is the set of all $x\; \backslash in\; X$ for which there exists a net (valued) in $S$ that converges to $x$ in $(X,\; \backslash tau).$

External links

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