In

topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

:
* In any space, $\backslash varnothing\; =\; \backslash operatorname\; \backslash varnothing$. In other words, the closure of the empty set $\backslash varnothing$ is $\backslash varnothing$ itself.
* 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 _{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, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

$(X,\; \backslash tau)$, 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

topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...

, the closure of a subset of points in a topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

consists of all points in together with all limit points of . The closure of may equivalently be defined as the union of and its boundary
Boundary or Boundaries may refer to:
* Border, in political geography
Entertainment
* ''Boundaries'' (2016 film), a 2016 Canadian film
* ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film
* Boundary (cricket), the edge of the pl ...

, and also as the intersection of all closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...

s containing . Intuitively, the closure can be thought of as all the points that are either in or "near" . A point which is in the closure of is a point of closure of . The notion of closure is in many ways dual to the notion of interior.
Definitions

Point of closure

For $S$ as a subset of aEuclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...

, $x$ is a point of closure of $S$ if every open ball
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defi ...

centered at $x$ contains a point of $S$ (this point can be $x$ itself).
This definition generalizes to any subset $S$ of a metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

$X.$ Fully expressed, for $X$ as 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 $d(x,\; s)\; <\; r$ ($x\; =\; s$ is allowed). Another way to express this is to say that $x$ is a point of closure of $S$ if the distance $d(x,\; S)\; :=\; \backslash inf\_\; d(x,\; s)\; =\; 0$ where $\backslash inf$ is the infimum.
This definition generalizes to topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

s by replacing "open ball" or "ball" with "neighbourhood
A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural area, ...

". 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 $S$ (again, $x\; =\; s$ for $s\; \backslash in\; S$ is allowed). 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 of a set. The difference between the two definitions is subtle but important – namely, in the definition of a limit point $x$ of a set $S$, every neighbourhood of $x$ must contain a point of $S$ . (Each neighbourhood of $x$ has $x$ but it also must have a point of $S$ that is different from $x$.) A limit point of $S$ has more strict condition than a point of closure of $S$ in the definitions. The set of all limit points of a set $S$ is called the . A limit point of a set is also called ''cluster point'' or ''accumulation point'' of the set. 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 there is a neighbourhood of $x$ which contains no other points of $S$ 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 atopological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...

$(X,\; \backslash tau),$ 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, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...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), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film * Boundary (cricket), the edge of the pl ...$\backslash partial(S).$
- $\backslash operatorname\; S$ is the set of all $x\; \backslash in\; X$ for which there exists a net (valued) in $S$ that converges to $x$ in $(X,\; \backslash tau).$

if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bico ...

$S\; =\; \backslash operatorname\; S$.
* 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 (such as a metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

), $\backslash operatorname\; S$ is the set of all limits of all convergent sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...

s of points in $S.$ For a general topological space, this statement remains true if one replaces "sequence" by " net" or " filter" (as described in the article on filters in topology).
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 operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S
:
Closure operators are de ...

below.
Examples

Consider asphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...

in a 3 dimensional space. Implicitly there are two regions of interest created by this sphere; the sphere itself and its interior (which is called an open 3-ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used ...

). It is useful to distinguish between the interior and the surface of the sphere, so we distinguish between the open 3-ball (the interior of the sphere), and the closed 3-ball – the closure of the open 3-ball that is the open 3-ball plus the surface (the surface as the sphere itself).
In real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...

s, then $\backslash operatorname\_X\; ((0,\; 1))\; =;\; href="/html/ALL/s/,\_1.html"\; ;"title=",\; 1">,\; 1$rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rati ...

s is the whole space $\backslash mathbb.$ We say that $\backslash mathbb$ is dense in $\backslash mathbb.$
* If $X$ is the complex plane
In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...

$\backslash mathbb\; =\; \backslash mathbb^2,$ 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 Tlower limit topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of inte ...

, 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'' from each other in a certain sense. The discrete topology is the finest to ...

, since every set is closed (and also open), every set is equal to its closure.
* In any indiscrete space In 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, concrete or codiscrete. Intuitively, this has the consequ ...

$X,$ 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 thepower set
In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is pos ...

of $X,$ $\backslash mathcal(X)$, into itself which satisfies the Kuratowski closure axioms.
Given a closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a clo ...

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, open sets are a generalization of open intervals in the real line.
In a metric space (a Set (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...

s 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 theempty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...

is the empty set;
* The closure of $X$ itself is $X.$
* The closure of an intersection of sets is always a subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

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
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

of the union of the closures.
** Thus, just as the union of two closed sets is closed, so too does closure distribute over binary unions: that is, $\backslash operatorname\_X\; (S\; \backslash cup\; T)\; =\; (\backslash operatorname\_X\; S)\; \backslash cup\; (\backslash operatorname\_X\; T).$ But just as a union of infinitely many closed sets is not necessarily closed, so too does closure not necessarily distribute over infinite unions: that is, $\backslash operatorname\_X\; \backslash left(\backslash bigcup\_\; S\_i\backslash right)\; \backslash neq\; \backslash bigcup\_\; \backslash operatorname\_X\; S\_i$ is possible when $I$ is infinite.
If $S\; \backslash subseteq\; T\; \backslash subseteq\; X$ and if $T$ is a 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 $\backslash operatorname\_T\; S$ 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 $T\; \backslash cap\; \backslash operatorname\_X\; S\; \backslash subseteq\; T\; \backslash cap\; C\; =\; \backslash operatorname\_T\; S.$ $\backslash blacksquare$
It follows that $S\; \backslash subseteq\; T$ is a dense subset of $T$ if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bico ...

$T$ is a subset of $\backslash operatorname\_X\; S.$
It is possible for $\backslash operatorname\_T\; S\; =\; T\; \backslash cap\; \backslash operatorname\_X\; S$ to be a proper subset of $\backslash operatorname\_X\; S;$ for example, take $X\; =\; \backslash R,$ $S\; =\; (0,\; 1),$ and $T\; =\; (0,\; \backslash infty).$
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 always guaranteed, 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 $T$ happens to an open subset of $X$ then the equality $\backslash operatorname\_T\; (S\; \backslash cap\; T)\; =\; T\; \backslash cap\; \backslash operatorname\_X\; S$ will hold (no matter the relationship between $S$ and $T$).
Let $S,\; T\; \backslash subseteq\; X$ 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 $C\; \backslash subseteq\; \backslash operatorname\_X\; (T\; \backslash cap\; S)\; \backslash subseteq\; \backslash operatorname\_X\; S.$ $\backslash blacksquare$
Consequently, if $\backslash mathcal$ is any open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\ ...

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
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...

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 in $X$", meaning that if $\backslash mathcal$ is any open cover
In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\ ...

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

Continuity

A function $f\; :\; X\; \backslash to\; Y$ between topological spaces is continuous if and only if thepreimage
In mathematics, the image of a function is the set of all output values it may produce.
More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through ...

of every closed subset of the codomain is closed in the domain; explicitly, this means: $f^(C)$ is closed in $X$ whenever $C$ is a closed subset of $Y.$
In terms of the closure operator, $f\; :\; X\; \backslash to\; Y$ is continuous if and only if for every subset $A\; \backslash subseteq\; X,$
$$f\backslash left(\backslash operatorname\_X\; A\backslash right)\; ~\backslash subseteq~\; \backslash operatorname\_Y\; (f(A)).$$
That is to say, given any element $x\; \backslash in\; X$ that belongs to the closure of a subset $A\; \backslash subseteq\; X,$ $f(x)$ necessarily belongs to the closure of $f(A)$ in $Y.$ If we declare that a point $x$ is a subset $A\; \backslash subseteq\; X$ if $x\; \backslash in\; \backslash operatorname\_X\; A,$ then this terminology allows for a plain English description of continuity: $f$ 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 $f(A).$ Thus continuous functions are exactly those functions that preserve (in the forward direction) the "closeness" relationship between points and sets: a function is continuous if and only if whenever a point is close to a set then the image of that point is close to the image of that set.
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 $f(A).$
Closed maps

A function $f\; :\; X\; \backslash to\; Y$ is a (strongly) closed map if and only if whenever $C$ is a closed subset of $X$ then $f(C)$ is a closed subset of $Y.$ In terms of the closure operator, $f\; :\; X\; \backslash to\; Y$ is a (strongly) closed map if and only if $\backslash operatorname\_Y\; f(A)\; \backslash subseteq\; f\backslash left(\backslash operatorname\_X\; A\backslash right)$ for every subset $A\; \backslash subseteq\; X.$ Equivalently, $f\; :\; X\; \backslash to\; Y$ is a (strongly) closed map if and only if $\backslash operatorname\_Y\; f(C)\; \backslash subseteq\; f(C)$ for every closed subset $C\; \backslash subseteq\; X.$Categorical interpretation

One may define the closure operator in terms of universal arrows, as follows. The powerset of a set $X$ may be realized as a partial ordercategory
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce)
...

$P$ 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 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, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics.
Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky ...

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

* * * Closed regular set, a set equal to the closure of their interior * * *Notes

References

Bibliography

* * * * * * * *External links

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