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In
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 a
Euclidean 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 $s \in S$ such that the distance $d\left(x, s\right) < r$ ($x = s$ is allowed). Another way to express this is to say that $x$ is a point of closure of $S$ if the distance $d\left(x, S\right) := \inf_ d\left(x, s\right) = 0$ where $\inf$ 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 $s \in S$ 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 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 ...
$\left(X, \tau\right),$ denoted by $\operatorname_ S$ or possibly by $\operatorname_X S$ (if $\tau$ is understood), where if both $X$ and $\tau$ are clear from context then it may also be denoted by $\operatorname S,$ $\overline,$ or $S ^$ (Moreover, $\operatorname$ is sometimes capitalized to $\operatorname$.) can be defined using any of the following equivalent definitions:
1. $\operatorname S$ is the set of all points of closure of $S.$
2. $\operatorname S$ is the set $S$ together with all of its limit points.
3. $\operatorname S$ 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 $S.$
4. $\operatorname S$ is the smallest closed set containing $S.$
5. $\operatorname S$ 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 ...
$\partial\left(S\right).$
6. $\operatorname S$ is the set of all $x \in X$ for which there exists a net (valued) in $S$ that converges to $x$ in $\left(X, \tau\right).$
The closure of a set has the following properties. * $\operatorname S$ is a closed superset of $S$. * The set $S$ is closed
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 ...
$S = \operatorname S$. * If $S \subseteq T$ then $\operatorname S$ is a subset of $\operatorname T.$ * If $A$ is a closed set, then $A$ contains $S$ if and only if $A$ contains $\operatorname S.$ 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 ...
), $\operatorname S$ 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 a
sphere 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
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, $\varnothing = \operatorname \varnothing$. In other words, the closure of the empty set $\varnothing$ is $\varnothing$ itself. * In any space $X,$ $X = \operatorname X.$ Giving $\mathbb$ and $\mathbb$ the standard (metric) topology: * If $X$ is the Euclidean space $\mathbb$ of
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
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 $\mathbb.$ We say that $\mathbb$ is dense in $\mathbb.$ * 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 ...
$\mathbb = \mathbb^2,$ then $\operatorname_X \left\left( \ \right\right) = \.$ * If $S$ is a finite subset of a Euclidean space $X,$ then $\operatorname_X S = S.$ (For a general topological space, this property is equivalent to the T1 axiom.) On the set of real numbers one can put other topologies rather than the standard one. * If $X = \mathbb$ is endowed with the
lower 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 * If one considers on $X = \mathbb$ the discrete topology in which every set is closed (open), then $\operatorname_X \left(\left(0, 1\right)\right) = \left(0, 1\right).$ * If one considers on $X = \mathbb$ the trivial topology in which the only closed (open) sets are the empty set and $\mathbb$ itself, then $\operatorname_X \left(\left(0, 1\right)\right) = \mathbb.$ 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 $X,$ $\operatorname_X A = X.$ 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 $\mathbb,$ and if $S = \,$ then $S$ is both closed and open in $\mathbb$ because neither $S$ nor its complement can contain $\sqrt2$, which would be the lower bound of $S$, but cannot be in $S$ because $\sqrt2$ is irrational. So, $S$ has no well defined closure due to boundary elements not being in $\mathbb$. 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 $\sqrt2$.

# Closure operator

A on a set $X$ is a mapping of the
power 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 $X,$ $\mathcal\left(X\right)$, into itself which satisfies the Kuratowski closure axioms. Given 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 ...
$\left(X, \tau\right)$, the topological closure induces a function $\operatorname_X : \wp\left(X\right) \to \wp\left(X\right)$ that is defined by sending a subset $S \subseteq X$ to $\operatorname_X S,$ where the notation $\overline$ or $S^$ may be used instead. Conversely, if $\mathbb$ is a closure operator on a set $X,$ then a topological space is obtained by defining the
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 $S \subseteq X$ that satisfy $\mathbb\left(S\right) = S$ (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 $\operatorname_X$ is dual to the interior operator, which is denoted by $\operatorname_X,$ in the sense that :$\operatorname_X S = X \setminus \operatorname_X \left(X \setminus S\right),$ and also :$\operatorname_X S = X \setminus \operatorname_X \left(X \setminus S\right).$ 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:

A subset $S$ is closed in $X$ if and only if $\operatorname_X S = S.$ In particular: * The closure of the
empty 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, $\operatorname_X \left(S \cup T\right) = \left(\operatorname_X S\right) \cup \left(\operatorname_X T\right).$ 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, $\operatorname_X \left\left(\bigcup_ S_i\right\right) \neq \bigcup_ \operatorname_X S_i$ is possible when $I$ is infinite. If $S \subseteq T \subseteq X$ and if $T$ is a subspace of $X$ (meaning that $T$ is endowed with the subspace topology that $X$ induces on it), then $\operatorname_T S \subseteq \operatorname_X S$ and the closure of $S$ computed in $T$ is equal to the intersection of $T$ and the closure of $S$ computed in $X$: $\operatorname_T S ~=~ T \cap \operatorname_X S.$ Because $\operatorname_X S$ is a closed subset of $X,$ the intersection $T \cap \operatorname_X S$ is a closed subset of $T$ (by definition of the subspace topology), which implies that $\operatorname_T S \subseteq T \cap \operatorname_X S$ (because $\operatorname_T S$ is the closed subset of $T$ containing $S$). Because $\operatorname_T S$ is a closed subset of $T,$ from the definition of the subspace topology, there must exist some set $C \subseteq X$ such that $C$ is closed in $X$ and $\operatorname_T S = T \cap C.$ Because $S \subseteq \operatorname_T S \subseteq C$ and $C$ is closed in $X,$ the minimality of $\operatorname_X S$ implies that $\operatorname_X S \subseteq C.$ Intersecting both sides with $T$ shows that $T \cap \operatorname_X S \subseteq T \cap C = \operatorname_T S.$ $\blacksquare$ It follows that $S \subseteq T$ 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 $\operatorname_X S.$ It is possible for $\operatorname_T S = T \cap \operatorname_X S$ to be a proper subset of $\operatorname_X S;$ for example, take $X = \R,$ $S = \left(0, 1\right),$ and $T = \left(0, \infty\right).$ If $S, T \subseteq X$ but $S$ is not necessarily a subset of $T$ then only $\operatorname_T (S \cap T) ~\subseteq~ T \cap \operatorname_X S$ is always guaranteed, where this containment could be strict (consider for instance $X = \R$ with the usual topology, $T = \left(-\infty, 0\right],$ and $S = \left(0, \infty\right)$From $T := \left(-\infty, 0\right]$ and $S := \left(0, \infty\right)$ it follows that $S \cap T = \varnothing$ and $\operatorname_X S = \left[0, \infty\right),$ which implies $\varnothing ~=~ \operatorname_T (S \cap T) ~\neq~ T \cap \operatorname_X S ~=~ \.$ ), although if $T$ happens to an open subset of $X$ then the equality $\operatorname_T \left(S \cap T\right) = T \cap \operatorname_X S$ will hold (no matter the relationship between $S$ and $T$). Let $S, T \subseteq X$ and assume that $T$ is open in $X.$ Let $C := \operatorname_T \left(T \cap S\right),$ which is equal to $T \cap \operatorname_X \left(T \cap S\right)$ (because $T \cap S \subseteq T \subseteq X$). The complement $T \setminus C$ is open in $T,$ where $T$ being open in $X$ now implies that $T \setminus C$ is also open in $X.$ Consequently $X \setminus \left(T \setminus C\right) = \left(X \setminus T\right) \cup C$ is a closed subset of $X$ where $\left(X \setminus T\right) \cup C$ contains $S$ as a subset (because if $s \in S$ is in $T$ then $s \in T \cap S \subseteq \operatorname_T \left(T \cap S\right) = C$), which implies that $\operatorname_X S \subseteq \left(X \setminus T\right) \cup C.$ Intersecting both sides with $T$ proves that $T \cap \operatorname_X S \subseteq T \cap C = C.$ The reverse inclusion follows from $C \subseteq \operatorname_X \left(T \cap S\right) \subseteq \operatorname_X S.$ $\blacksquare$ Consequently, if $\mathcal$ 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 $S \subseteq X$ is any subset then: $\operatorname_X S = \bigcup_ \operatorname_U (U \cap S)$ because $\operatorname_U \left(S \cap U\right) = U \cap \operatorname_X S$ for every $U \in \mathcal$ (where every $U \in \mathcal$ 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 $\mathcal$ are domains of coordinate charts. In words, this result shows that the closure in $X$ of any subset $S \subseteq X$ 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 $S \subseteq X$ is closed in $X$ if and only if it is " locally closed in $X$", meaning that if $\mathcal$ 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 $S \cap U$ is closed in $U$ for every $U \in \mathcal.$

# Functions and closure

## Continuity

A function $f : X \to Y$ between topological spaces is continuous if and only if the
preimage 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: $f^\left(C\right)$ is closed in $X$ whenever $C$ is a closed subset of $Y.$ In terms of the closure operator, $f : X \to Y$ is continuous if and only if for every subset $A \subseteq X,$ $f\left(\operatorname_X A\right) ~\subseteq~ \operatorname_Y (f(A)).$ That is to say, given any element $x \in X$ that belongs to the closure of a subset $A \subseteq X,$ $f\left(x\right)$ necessarily belongs to the closure of $f\left(A\right)$ in $Y.$ If we declare that a point $x$ is a subset $A \subseteq X$ if $x \in \operatorname_X A,$ then this terminology allows for a plain English description of continuity: $f$ is continuous if and only if for every subset $A \subseteq X,$ $f$ maps points that are close to $A$ to points that are close to $f\left(A\right).$ 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 $x \in X$ if and only if whenever $x$ is close to a subset $A \subseteq X,$ then $f\left(x\right)$ is close to $f\left(A\right).$

## Closed maps

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

# 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 order
category 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 $A \to B$ whenever $A$ is a subset of $B.$ Furthermore, a topology $T$ on $X$ is a subcategory of $P$ with inclusion functor $I : T \to P.$ The set of closed subsets containing a fixed subset $A \subseteq X$ can be identified with the comma category $\left(A \downarrow I\right).$ This category — also a partial order — then has initial object $\operatorname A.$ Thus there is a universal arrow from $A$ to $I,$ given by the inclusion $A \to \operatorname A.$ Similarly, since every closed set containing $X \setminus A$ corresponds with an open set contained in $A$ we can interpret the category $\left(I \downarrow X \setminus A\right)$ as the set of open subsets contained in $A,$ with terminal object $\operatorname\left(A\right),$ 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.

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

* * * * * * * *