Intopology, theclosure of a subsetS of points in atopological space consists of allpoints inS together with alllimit points ofS. The closure ofS may equivalently be defined as theunion ofS and itsboundary, and also as theintersection of allclosed sets containingS. Intuitively, the closure can be thought of as all the points that are either inS or "very near"S. A point which is in the closure ofS is apoint of closure ofS. The notion of closure is in many waysdual to the notion ofinterior.

For${\displaystyle S}$  as a subset of aEuclidean space,${\displaystyle x}$  is a point of closure of${\displaystyle S}$  if everyopen ball centered at${\displaystyle x}$  contains a point of${\displaystyle S}$  (this point can be${\displaystyle x}$  itself).

This definition generalizes to any subset${\displaystyle S}$  of ametric space${\displaystyle X.}$  Fully expressed, for${\displaystyle X}$  as a metric space with metric${\displaystyle d,}$ ${\displaystyle x}$  is a point of closure of${\displaystyle S}$  if for every${\displaystyle r>0}$  there exists some${\displaystyle s\in S}$  such that the distance  (${\displaystyle x=s}$  is allowed). Another way to express this is to say that${\displaystyle x}$  is a point of closure of${\displaystyle S}$  if the distance${\displaystyle d(x,S):=\underset{s\in S}{inf}d(x,s)=0}$  where${\displaystyle inf}$  is theinfimum.

This definition generalizes totopological spaces by replacing "open ball" or "ball" with "neighbourhood". Let${\displaystyle S}$  be a subset of a topological space${\displaystyle X.}$  Then${\displaystyle x}$  is apoint of closure oradherent point of${\displaystyle S}$  if every neighbourhood of${\displaystyle x}$  contains a point of${\displaystyle S}$  (again,${\displaystyle x=s}$  for${\displaystyle s\in S}$  is allowed).^[1] Note that this definition does not depend upon whether neighbourhoods are required to be open.

The definition of a point of closure of a set is closely related to the definition of alimit point of a set. The difference between the two definitions is subtle but important – namely, in the definition of a limit point${\displaystyle x}$  of a set${\displaystyle S}$ , every neighbourhood of${\displaystyle x}$  must contain a point of${\displaystyle S}$ other than${\displaystyle x}$  itself, i.e., each neighbourhood of${\displaystyle x}$  obviously has${\displaystyle x}$  but it also must have a point of${\displaystyle S}$  that is not equal to${\displaystyle x}$  in order for${\displaystyle x}$  to be a limit point of${\displaystyle S}$ . A limit point of${\displaystyle S}$  has more strict condition than a point of closure of${\displaystyle S}$  in the definitions. The set of all limit points of a set${\displaystyle S}$  is called thederived set of${\displaystyle S}$ . A limit point of a set is also calledcluster point oraccumulation 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 anisolated point. In other words, a point${\displaystyle x}$  is an isolated point of${\displaystyle S}$  if it is an element of${\displaystyle S}$  and there is a neighbourhood of${\displaystyle x}$  which contains no other points of${\displaystyle S}$  than${\displaystyle x}$  itself.^[2]

For a given set${\displaystyle S}$  and point${\displaystyle x,}$ ${\displaystyle x}$  is a point of closure of${\displaystyle S}$  if and only if${\displaystyle x}$  is an element of${\displaystyle S}$  or${\displaystyle x}$  is a limit point of${\displaystyle S}$  (or both).

Theclosure of a subset${\displaystyle S}$  of atopological space${\displaystyle (X,\tau ),}$  denoted by${\displaystyle {\mathrm{cl}}_{(X,\tau )}S}$  or possibly by${\displaystyle {\mathrm{cl}}_{X}S}$  (if${\displaystyle \tau }$  is understood), where if both${\displaystyle X}$  and${\displaystyle \tau }$  are clear from context then it may also be denoted by${\displaystyle \mathrm{cl}S,}$ ${\displaystyle \overline{S},}$  or${\displaystyle S{}^{-}}$  (Moreover,${\displaystyle \mathrm{cl}}$  is sometimes capitalized to${\displaystyle \mathrm{Cl}}$ .) can be defined using any of the following equivalent definitions:

1. ${\displaystyle \mathrm{cl}S}$  is the set of allpoints of closure of${\displaystyle S.}$ 
2. ${\displaystyle \mathrm{cl}S}$  is the set${\displaystyle S}$  together withall of its limit points. (Each point of${\displaystyle S}$  is a point of closure of${\displaystyle S}$ , and each limit point of${\displaystyle S}$  is also a point of closure of${\displaystyle S}$ .)^[3]
3. ${\displaystyle \mathrm{cl}S}$  is the intersection of allclosed sets containing${\displaystyle S.}$ 
4. ${\displaystyle \mathrm{cl}S}$  is the smallest closed set containing${\displaystyle S.}$ 
5. ${\displaystyle \mathrm{cl}S}$  is the union of${\displaystyle S}$  and itsboundary${\displaystyle \mathrm{\partial }(S).}$ 
6. ${\displaystyle \mathrm{cl}S}$  is the set of all${\displaystyle x\in X}$  for which there exists anet (valued) in${\displaystyle S}$  that converges to${\displaystyle x}$  in${\displaystyle (X,\tau ).}$ 

The closure of a set has the following properties.^[4]

• ${\displaystyle \mathrm{cl}S}$  is aclosed superset of${\displaystyle S}$ .
• The set${\displaystyle S}$  is closedif and only if${\displaystyle S=\mathrm{cl}S}$ .
• If${\displaystyle S\subseteq T}$  then${\displaystyle \mathrm{cl}S}$  is a subset of${\displaystyle \mathrm{cl}T.}$ 
• If${\displaystyle A}$  is a closed set, then${\displaystyle A}$  contains${\displaystyle S}$  if and only if${\displaystyle A}$  contains${\displaystyle \mathrm{cl}S.}$ 

Sometimes the second or third property above is taken as thedefinition of the topological closure, which still make sense when applied to other types of closures (see below).^[5]

In afirst-countable space (such as ametric space),${\displaystyle \mathrm{cl}S}$  is the set of alllimits of all convergentsequences of points in${\displaystyle S.}$  For a general topological space, this statement remains true if one replaces "sequence" by "net" or "filter" (as described in the article onfilters 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, seeclosure operator below.

Consider asphere 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). 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).

Intopological space:

• In any space,${\displaystyle \varnothing =\mathrm{cl}\varnothing }$ . In other words, the closure of the empty set${\displaystyle \varnothing }$  is${\displaystyle \varnothing }$  itself.
• In any space${\displaystyle X,}$ ${\displaystyle X=\mathrm{cl}X.}$ 

Giving${\displaystyle \mathbb{R}}$  and${\displaystyle \mathbb{C}}$  thestandard (metric) topology:

• If${\displaystyle X}$  is the Euclidean space${\displaystyle \mathbb{R}}$  ofreal numbers, then${\displaystyle {\mathrm{cl}}_X((0,1))=[0,1]}$ . In other words., the closure of the set${\displaystyle (0,1)}$  as a subset of${\displaystyle X}$  is${\displaystyle [0,1]}$ .
• If${\displaystyle X}$  is the Euclidean space${\displaystyle \mathbb{R}}$ , then the closure of the set${\displaystyle \mathbb{Q}}$  ofrational numbers is the whole space${\displaystyle \mathbb{R}.}$  We say that${\displaystyle \mathbb{Q}}$  isdense in${\displaystyle \mathbb{R}.}$ 
• If${\displaystyle X}$  is thecomplex plane${\displaystyle \mathbb{C}={\mathbb{R}}^2,}$  then${\displaystyle {\mathrm{cl}}_X\left(\{z\in \mathbb{C}:|z|>1\}\right)=\{z\in \mathbb{C}:|z|\ge 1\}.}$ 
• If${\displaystyle S}$  is afinite subset of a Euclidean space${\displaystyle X,}$  then${\displaystyle {\mathrm{cl}}_{X}S=S.}$  (For a general topological space, this property is equivalent to theT₁ axiom.)

On the set of real numbers one can put other topologies rather than the standard one.

• If${\displaystyle X=\mathbb{R}}$  is endowed with thelower limit topology, then${\displaystyle {\mathrm{cl}}_X((0,1))=[0,1).}$ 
• If one considers on${\displaystyle X=\mathbb{R}}$  thediscrete topology in which every set is closed (open), then${\displaystyle {\mathrm{cl}}_X((0,1))=(0,1).}$ 
• If one considers on${\displaystyle X=\mathbb{R}}$  thetrivial topology in which the only closed (open) sets are the empty set and${\displaystyle \mathbb{R}}$  itself, then${\displaystyle {\mathrm{cl}}_X((0,1))=\mathbb{R}.}$ 

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 anydiscrete space, since every set is closed (and also open), every set is equal to its closure.
• In anyindiscrete space${\displaystyle X,}$  since the only closed sets are the empty set and${\displaystyle X}$  itself, we have that the closure of the empty set is the empty set, and for every non-empty subset${\displaystyle A}$  of${\displaystyle X,}$ ${\displaystyle {\mathrm{cl}}_{X}A=X.}$  In other words, every non-empty subset of an indiscrete space isdense.

The closure of a set also depends upon in which space we are taking the closure. For example, if${\displaystyle X}$  is the set of rational numbers, with the usualrelative topology induced by the Euclidean space${\displaystyle \mathbb{R},}$  and if${\displaystyle S=\{q\in \mathbb{Q}:q^2>2,q>0\},}$  then${\displaystyle S}$  isboth closed and open in${\displaystyle \mathbb{Q}}$  because neither${\displaystyle S}$  nor its complement can contain${\displaystyle \sqrt{2}}$ , which would be the lower bound of${\displaystyle S}$ , but cannot be in${\displaystyle S}$  because${\displaystyle \sqrt{2}}$  is irrational. So,${\displaystyle S}$  has no well defined closure due to boundary elements not being in${\displaystyle \mathbb{Q}}$ . However, if we instead define${\displaystyle X}$  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 allreal numbers greater thanor equal to${\displaystyle \sqrt{2}}$ .

Aclosure operator on a set${\displaystyle X}$  is amapping of thepower set of${\displaystyle X,}$ ${\displaystyle \mathcal{P}(X)}$ , into itself which satisfies theKuratowski closure axioms. Given atopological space${\displaystyle (X,\tau )}$ , the topological closure induces a function${\displaystyle {\mathrm{cl}}_X:\mathrm{\wp }(X)\to \mathrm{\wp }(X)}$  that is defined by sending a subset${\displaystyle S\subseteq X}$  to${\displaystyle {\mathrm{cl}}_{X}S,}$  where the notation${\displaystyle \overline{S}}$  or${\displaystyle S^{-}}$  may be used instead. Conversely, if${\displaystyle \mathbb{c}}$  is a closure operator on a set${\displaystyle X,}$  then a topological space is obtained by defining theclosed sets as being exactly those subsets${\displaystyle S\subseteq X}$  that satisfy${\displaystyle \mathbb{c}(S)=S}$  (so complements in${\displaystyle X}$  of these subsets form theopen sets of the topology).^[6]

The closure operator${\displaystyle {\mathrm{cl}}_X}$  isdual to theinterior operator, which is denoted by${\displaystyle {\mathrm{int}}_X,}$  in the sense that

${\displaystyle {\mathrm{cl}}_{X}S=X\setminus {\mathrm{int}}_X(X\setminus S),}$ 

and also

${\displaystyle {\mathrm{int}}_{X}S=X\setminus {\mathrm{cl}}_X(X\setminus S).}$ 

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 theircomplements in${\displaystyle X.}$ 

In general, the closure operator does not commute with intersections. However, in acomplete metric space the following result does hold:

A subset${\displaystyle S}$  isclosed in${\displaystyle X}$  if and only if${\displaystyle {\mathrm{cl}}_{X}S=S.}$  In particular:

• The closure of theempty set is the empty set;
• The closure of${\displaystyle X}$  itself is${\displaystyle X.}$ 
• The closure of anintersection of sets is always asubset of (but need not be equal to) the intersection of the closures of the sets.
• In aunion offinitely 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 asuperset 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,${\displaystyle {\mathrm{cl}}_X(S\cup T)=({\mathrm{cl}}_{X}S)\cup ({\mathrm{cl}}_{X}T).}$  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,${\displaystyle {\mathrm{cl}}_X\left(\bigcup _{i\in I}{S}_i\right)\ne \bigcup _{i\in I}{\mathrm{cl}}_X{S}_i}$  is possible when${\displaystyle I}$  is infinite.

If${\displaystyle S\subseteq T\subseteq X}$  and if${\displaystyle T}$  is asubspace of${\displaystyle X}$  (meaning that${\displaystyle T}$  is endowed with thesubspace topology that${\displaystyle X}$  induces on it), then${\displaystyle {\mathrm{cl}}_{T}S\subseteq {\mathrm{cl}}_{X}S}$  and the closure of${\displaystyle S}$  computed in${\displaystyle T}$  is equal to the intersection of${\displaystyle T}$  and the closure of${\displaystyle S}$  computed in${\displaystyle X}$ :$${\displaystyle {\mathrm{cl}}_{T}S=T\cap {\mathrm{cl}}_{X}S.}$$ 

It follows that${\displaystyle S\subseteq T}$  is a dense subset of${\displaystyle T}$ if and only if${\displaystyle T}$  is a subset of${\displaystyle {\mathrm{cl}}_{X}S.}$  It is possible for${\displaystyle {\mathrm{cl}}_{T}S=T\cap {\mathrm{cl}}_{X}S}$  to be a proper subset of${\displaystyle {\mathrm{cl}}_{X}S;}$  for example, take${\displaystyle X=\mathbb{R},}$ ${\displaystyle S=(0,1),}$  and${\displaystyle T=(0,\mathrm{\infty }).}$ 

If${\displaystyle S,T\subseteq X}$  but${\displaystyle S}$  is not necessarily a subset of${\displaystyle T}$  then only$${\displaystyle {\mathrm{cl}}_T(S\cap T)\subseteq T\cap {\mathrm{cl}}_{X}S}$$ is always guaranteed, where this containment could be strict (consider for instance${\displaystyle X=\mathbb{R}}$  with the usual topology,${\displaystyle T=(-\mathrm{\infty },0],}$  and${\displaystyle S=(0,\mathrm{\infty })}$ ^{[proof 1]}), although if${\displaystyle T}$  happens to an open subset of${\displaystyle X}$  then the equality${\displaystyle {\mathrm{cl}}_T(S\cap T)=T\cap {\mathrm{cl}}_{X}S}$  will hold (no matter the relationship between${\displaystyle S}$  and${\displaystyle T}$ ).

Consequently, if${\displaystyle \mathcal{U}}$  is anyopen cover of${\displaystyle X}$  and if${\displaystyle S\subseteq X}$  is any subset then:$${\displaystyle {\mathrm{cl}}_{X}S=\bigcup _{U\in \mathcal{U}}{\mathrm{cl}}_U(U\cap S)}$$  because${\displaystyle {\mathrm{cl}}_U(S\cap U)=U\cap {\mathrm{cl}}_{X}S}$  for every${\displaystyle U\in \mathcal{U}}$  (where every${\displaystyle U\in \mathcal{U}}$  is endowed with thesubspace topology induced on it by${\displaystyle X}$ ). This equality is particularly useful when${\displaystyle X}$  is amanifold and the sets in the open cover${\displaystyle \mathcal{U}}$  are domains ofcoordinate charts. In words, this result shows that the closure in${\displaystyle X}$  of any subset${\displaystyle S\subseteq X}$  can be computed "locally" in the sets of any open cover of${\displaystyle X}$  and then unioned together.In this way, this result can be viewed as the analogue of the well-known fact that a subset${\displaystyle S\subseteq X}$  is closed in${\displaystyle X}$  if and only if it is "locally closed in${\displaystyle X}$ ", meaning that if${\displaystyle \mathcal{U}}$  is anyopen cover of${\displaystyle X}$  then${\displaystyle S}$  is closed in${\displaystyle X}$  if and only if${\displaystyle S\cap U}$  is closed in${\displaystyle U}$  for every${\displaystyle U\in \mathcal{U}.}$ 

A function${\displaystyle f:X\to Y}$  between topological spaces iscontinuous if and only if thepreimage of every closed subset of the codomain is closed in the domain; explicitly, this means:${\displaystyle f^{-1}(C)}$  is closed in${\displaystyle X}$  whenever${\displaystyle C}$  is a closed subset of${\displaystyle Y.}$ 

In terms of the closure operator,${\displaystyle f:X\to Y}$  is continuous if and only if for every subset${\displaystyle A\subseteq X,}$ $${\displaystyle f\left({\mathrm{cl}}_{X}A\right)\subseteq {\mathrm{cl}}_Y(f(A)).}$$ That is to say, given any element${\displaystyle x\in X}$  that belongs to the closure of a subset${\displaystyle A\subseteq X,}$ ${\displaystyle f(x)}$  necessarily belongs to the closure of${\displaystyle f(A)}$  in${\displaystyle Y.}$  If we declare that a point${\displaystyle x}$  isclose to a subset${\displaystyle A\subseteq X}$  if${\displaystyle x\in {\mathrm{cl}}_{X}A,}$  then this terminology allows for aplain English description of continuity:${\displaystyle f}$  is continuous if and only if for every subset${\displaystyle A\subseteq X,}$ ${\displaystyle f}$  maps points that are close to${\displaystyle A}$  to points that are close to${\displaystyle f(A).}$  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,${\displaystyle f}$  is continuous at a fixed given point${\displaystyle x\in X}$  if and only if whenever${\displaystyle x}$  is close to a subset${\displaystyle A\subseteq X,}$  then${\displaystyle f(x)}$  is close to${\displaystyle f(A).}$ 

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

One may define the closure operator in terms of universal arrows, as follows.

Thepowerset of a set${\displaystyle X}$  may be realized as apartial ordercategory${\displaystyle P}$  in which the objects are subsets and the morphisms areinclusion maps${\displaystyle A\to B}$  whenever${\displaystyle A}$  is a subset of${\displaystyle B.}$  Furthermore, a topology${\displaystyle T}$  on${\displaystyle X}$  is asubcategory of${\displaystyle P}$  with inclusion functor${\displaystyle I:T\to P.}$  The set of closed subsets containing a fixed subset${\displaystyle A\subseteq X}$  can be identified with thecomma category${\displaystyle (A\downarrow I).}$  This category — also a partial order — then has initial object${\displaystyle \mathrm{cl}A.}$  Thus there is a universal arrow from${\displaystyle A}$  to${\displaystyle I,}$  given by the inclusion${\displaystyle A\to \mathrm{cl}A.}$ 

Similarly, since every closed set containing${\displaystyle X\setminus A}$  corresponds with an open set contained in${\displaystyle A}$  we can interpret the category${\displaystyle (I\downarrow X\setminus A)}$  as the set of open subsets contained in${\displaystyle A,}$  withterminal object${\displaystyle \mathrm{int}(A),}$  theinterior of${\displaystyle A.}$ 

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 examplealgebraic closure), since all are examples ofuniversal arrows.

• Adherent point – Point that belongs to the closure of some given subset of a topological space
• Closure algebra – Algebraic structurePages displaying short descriptions of redirect targets
• Closed regular set, a set equal to the closure of their interior
• Derived set (mathematics) – Set of all limit points of a set
• Interior (topology) – Largest open subset of some given set
• Limit point of a set – Cluster point in a topological spacePages displaying short descriptions of redirect targets

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