View source for Closure (topology)

{{short description|All points and limit points in a subset of a topological space}}
{{Other uses|Closure (disambiguation)}}

In [[topology]], the '''closure''' of a subset {{mvar|S}} of points in a [[topological space]] consists of all [[Topology glossary#P|point]]s in {{mvar|S}} together with all [[limit points]] of {{mvar|S}}. The closure of {{mvar|S}} may equivalently be defined as the [[Union (set theory)|union]] of {{mvar|S}} and its [[Boundary (topology)|boundary]], and also as the [[Intersection (set theory)|intersection]] of all [[closed set]]s containing {{mvar|S}}. Intuitively, the closure can be thought of as all the points that are either in {{mvar|S}} or "very near" {{mvar|S}}. A point which is in the closure of {{mvar|S}} is a [[adherent point|point of closure]] of {{mvar|S}}. The notion of closure is in many ways [[duality (mathematics)|dual]] to the notion of [[interior (topology)|interior]].

==Definitions==

=== Point of closure ===
{{Main|Adherent point}}

For <math>S</math> as a subset of a [[Euclidean space]], <math>x</math> is a point of closure of <math>S</math> if every [[open ball]] centered at <math>x</math> contains a point of <math>S</math> (this point can be <math>x</math> itself).

This definition generalizes to any subset <math>S</math> of a [[metric space]] <math>X.</math> Fully expressed, for <math>X</math> as a metric space with metric <math>d,</math> <math>x</math> is a point of closure of <math>S</math> if for every <math>r > 0</math> there exists some <math>s \in S</math> such that the distance <math>d(x, s) < r</math> (<math>x = s</math> is allowed). Another way to express this is to say that <math>x</math> is a point of closure of <math>S</math> if the distance <math>d(x, S) := \inf_{s \in S} d(x, s) = 0</math> where <math>\inf</math> is the [[Infimum and supremum|infimum]]. 

This definition generalizes to [[topological space]]s by replacing "open ball" or "ball" with "[[Topology glossary#N|neighbourhood]]". Let <math>S</math> be a subset of a topological space <math>X.</math> Then <math>x</math> is a {{em|'''[[Adherent point|point of closure]]'''}} or {{em|'''adherent point'''}} of <math>S</math> if every neighbourhood of <math>x</math> contains a point of <math>S</math> (again, <math>x = s</math> for <math>s \in S</math> is allowed).<ref>{{harvnb|Schubert|1968|loc=p. 20}}</ref> Note that this definition does not depend upon whether neighbourhoods are required to be open.

=== Limit point ===
{{Main|Limit point of a set}}

The definition of a point of closure of a set 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 <math>x</math> of a set <math>S</math>, every neighbourhood of <math>x</math> must contain a point of <math>S</math> {{em|other than <math>x</math> itself}}, i.e., each neighbourhood of <math>x</math> obviously has <math>x</math> but it also must have a point of <math>S</math> that is not equal to <math>x</math> in order for <math>x</math> to be a limit point of <math>S</math>. A limit point of <math>S</math> has more strict condition than a point of closure of <math>S</math> in the definitions. The set of all limit points of a set <math>S</math> is called the {{em|'''[[Derived set (mathematics)|derived set]]''' of <math>S</math>}}. 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 <math>x</math> is an isolated point of <math>S</math> if it is an element of <math>S</math> and there is a neighbourhood of <math>x</math> which contains no other points of <math>S</math> than <math>x</math> itself.<ref>{{harvnb|Kuratowski|1966|loc=p. 75}}</ref>

For a given set <math>S</math> and point <math>x,</math> <math>x</math> is a point of closure of <math>S</math> if and only if <math>x</math> is an element of <math>S</math> or <math>x</math> is a limit point of <math>S</math> (or both).

===Closure of a set===
{{See also|Closure (mathematics)}}

The {{em|'''closure'''}} of a subset <math>S</math> of a [[topological space]] <math>(X, \tau),</math> denoted by <math>\operatorname{cl}_{(X, \tau)} S</math> or possibly by <math>\operatorname{cl}_X S</math> (if <math>\tau</math> is understood), where if both <math>X</math> and <math>\tau</math> are clear from context then it may also be denoted by <math>\operatorname{cl} S,</math> <math>\overline{S},</math> or <math>S {}^{-}</math> (Moreover, <math>\operatorname{cl}</math> is sometimes capitalized to <math>\operatorname{Cl}</math>.) can be defined using any of the following equivalent definitions:
# <math>\operatorname{cl} S</math> is the set of all [[Adherent point|points of closure]] of <math>S.</math>
# <math>\operatorname{cl} S</math> is the set <math>S</math> together with [[Derived set (mathematics)|all of its limit points]]. (Each point of <math>S</math> is a point of closure of <math>S</math>, and each limit point of <math>S</math> is also a point of closure of <math>S</math>.)<ref>{{harvnb|Hocking|Young|1988|loc=p. 4}}</ref>
# <math>\operatorname{cl} S</math> is the intersection of all [[closed set]]s containing <math>S.</math>
# <math>\operatorname{cl} S</math> is the smallest closed set containing <math>S.</math>
# <math>\operatorname{cl} S</math> is the union of <math>S</math> and its [[Boundary (topology)|boundary]] <math>\partial(S).</math>
# <math>\operatorname{cl} S</math> is the set of all <math>x \in X</math> for which there exists a [[Net (mathematics)|net]] (valued) in <math>S</math> that converges to <math>x</math> in <math>(X, \tau).</math>

The closure of a set has the following properties.<ref>{{harvnb|Croom|1989|loc=p. 104}}</ref>

* <math>\operatorname{cl} S</math> is a [[closed set|closed]] superset of <math>S</math>.
* The set <math>S</math> is closed [[if and only if]] <math>S = \operatorname{cl} S</math>.
* If <math>S \subseteq T</math> then <math>\operatorname{cl} S</math> is a subset of <math>\operatorname{cl} T.</math>
* If <math>A</math> is a closed set, then <math>A</math> contains <math>S</math> if and only if <math>A</math> contains <math>\operatorname{cl} S.</math>

Sometimes the second or third property above is taken as the {{em|definition}} of the topological closure, which still make sense when applied to other types of closures (see below).<ref>{{harvnb|Gemignani|1990|loc=p. 55}}, {{harvnb|Pervin|1965|loc=p. 40}} and {{harvnb|Baker|1991|loc=p. 38}} use the second property as the definition.</ref>

In a [[first-countable space]] (such as a [[metric space]]), <math>\operatorname{cl} S</math> is the set of all [[Limit of a sequence|limits]] of all convergent [[sequence]]s of points in <math>S.</math> For a general topological space, this statement remains true if one replaces "sequence" by "[[Net (mathematics)|net]]" or "[[Filter (set theory)|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|closure operator]] below.

==Examples==
Consider a [[sphere]] 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 (mathematics)|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).

In [[topological space]]:
* In any space, <math>\varnothing = \operatorname{cl} \varnothing</math>. In other words, the closure of the empty set <math>\varnothing</math> is <math>\varnothing</math> itself. 
* In any space <math>X,</math> <math>X = \operatorname{cl} X.</math>

Giving <math>\mathbb{R}</math> and <math>\mathbb{C}</math> the [[Standard topology|standard (metric) topology]]:
* If <math>X</math> is the Euclidean space <math>\mathbb{R}</math> of [[real number]]s, then <math>\operatorname{cl}_X ((0, 1)) = [0, 1]</math>. In other words., the closure of the set <math>(0,1)</math> as a subset of <math>X</math> is <math>[0,1]</math>.
* If <math>X</math> is the Euclidean space <math>\mathbb{R}</math>, then the closure of the set <math>\mathbb{Q}</math> of [[rational number]]s is the whole space <math>\mathbb{R}.</math> We say that <math>\mathbb{Q}</math> is [[dense (topology)|dense]] in <math>\mathbb{R}.</math>
* If <math>X</math> is the [[complex number|complex plane]] <math>\mathbb{C} = \mathbb{R}^2,</math> then <math>\operatorname{cl}_X \left( \{ z \in \mathbb{C} : | z | > 1 \} \right) = \{ z \in \mathbb{C} : | z | \geq 1 \}.</math>
* If <math>S</math> is a [[finite set|finite]] subset of a Euclidean space <math>X,</math> then <math>\operatorname{cl}_X S = S.</math> (For a general topological space, this property is equivalent to the [[T1 space|T<sub>1</sub> axiom]].)

On the set of real numbers one can put other topologies rather than the standard one.
* If <math>X = \mathbb{R}</math> is endowed with the [[lower limit topology]], then <math>\operatorname{cl}_X ((0, 1)) = [0, 1).</math>
* If one considers on <math>X = \mathbb{R}</math> the [[discrete topology]] in which every set is closed (open), then <math>\operatorname{cl}_X ((0, 1)) = (0, 1).</math>
* If one considers on <math>X = \mathbb{R}</math> the [[trivial topology]] in which the only closed (open) sets are the empty set and <math>\mathbb{R}</math> itself, then <math>\operatorname{cl}_X ((0, 1)) = \mathbb{R}.</math>

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]] <math>X,</math> since the only closed sets are the empty set and <math>X</math> itself, we have that the closure of the empty set is the empty set, and for every non-empty subset <math>A</math> of <math>X,</math> <math>\operatorname{cl}_X A = X.</math> 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 <math>X</math> is the set of rational numbers, with the usual [[subspace topology|relative topology]] induced by the Euclidean space <math>\mathbb{R},</math> and if <math>S = \{ q \in \mathbb{Q} : q^2 > 2, q > 0 \},</math> then <math>S</math> is [[Clopen set|both closed and open]] in <math>\mathbb{Q}</math> because neither <math>S</math> nor its complement can contain <math>\sqrt2</math>, which would be the lower bound of <math>S</math>, but cannot be in <math>S</math> because <math>\sqrt2</math> is irrational. So, <math>S</math> has no well defined closure due to boundary elements not being in <math>\mathbb{Q}</math>. However, if we instead define <math>X</math> 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 {{em|real numbers}} greater than {{em|or equal to}} <math>\sqrt2</math>.

==Closure operator==
<!-- This section is linked from [[Closure (topology)]] -->
{{See also|Closure operator|Kuratowski closure axioms}}

A {{em|'''closure operator'''}} on a set <math>X</math> is a [[map (mathematics)|mapping]] of the [[power set]] of <math>X,</math> <math>\mathcal{P}(X)</math>, into itself which satisfies the [[Kuratowski closure axioms]]. 
Given a [[topological space]] <math>(X, \tau)</math>, the topological closure induces a function <math>\operatorname{cl}_X : \wp(X) \to \wp(X)</math> that is defined by sending a subset <math>S \subseteq X</math> to <math>\operatorname{cl}_X S,</math> where the notation <math>\overline{S}</math> or <math>S^{-}</math> may be used instead.  Conversely, if <math>\mathbb{c}</math> is a closure operator on a set <math>X,</math> then a topological space is obtained by defining the [[closed set]]s as being exactly those subsets <math>S \subseteq X</math> that satisfy <math>\mathbb{c}(S) = S</math> (so complements in <math>X</math> of these subsets form the [[open set]]s of the topology).<ref>{{harvnb|Pervin|1965|loc=p. 41}}</ref>

The closure operator <math>\operatorname{cl}_X</math> is [[Duality (mathematics)|dual]] to the [[Interior (topology)|interior]] operator, which is denoted by <math>\operatorname{int}_X,</math> in the sense that

:<math>\operatorname{cl}_X S = X \setminus \operatorname{int}_X (X \setminus S),</math>

and also

:<math>\operatorname{int}_X S = X \setminus \operatorname{cl}_X (X \setminus S).</math>

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 <math>X.</math>

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

{{Math theorem|name=Theorem{{sfn|Zălinescu|2002|p=33}}|note=C. Ursescu|math_statement=
Let <math>S_1, S_2, \ldots</math> be a sequence of subsets of a [[complete metric space]] <math>X.</math>
*If each <math>S_i</math> is closed in <math>X</math> then <math display="block">\operatorname{cl}_X \left( \bigcup_{i \in \N} \operatorname{int}_X S_i \right) = \operatorname{cl}_X \left[ \operatorname{int}_X \left( \bigcup_{i \in \N} S_i \right) \right].</math>
*If each <math>S_i</math> is open in <math>X</math> then <math display="block">\operatorname{int}_X \left( \bigcap_{i \in \N} \operatorname{cl}_X S_i \right) = \operatorname{int}_X \left[ \operatorname{cl}_X \left( \bigcap_{i \in \N} S_i \right) \right].</math>
}}

==Facts about closures==

A subset <math>S</math> is [[Closed set|closed]] in <math>X</math> if and only if <math>\operatorname{cl}_X S = S.</math> In particular:
* The closure of the [[empty set]] is the empty set;
* The closure of <math>X</math> itself is <math>X.</math>
* The closure of an [[Intersection (set theory)|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 (set theory)|union]] of [[Finite set|finite]]ly 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.
** Thus, just as the union of two closed sets is closed, so too does closure distribute over binary unions: that is, <math>\operatorname{cl}_X (S \cup T) = (\operatorname{cl}_X S) \cup (\operatorname{cl}_X T).</math> 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, <math>\operatorname{cl}_X \left(\bigcup_{i \in I} S_i\right) \neq \bigcup_{i \in I} \operatorname{cl}_X S_i</math> is possible when <math>I</math> is infinite.

If <math>S \subseteq T \subseteq X</math> and if <math>T</math> is a [[Topological subspace|subspace]] of <math>X</math> (meaning that <math>T</math> is endowed with the [[subspace topology]] that <math>X</math> induces on it), then <math>\operatorname{cl}_T S \subseteq \operatorname{cl}_X S</math> and the closure of <math>S</math> computed in <math>T</math> is equal to the intersection of <math>T</math> and the closure of <math>S</math> computed in <math>X</math>: 
<math display=block>\operatorname{cl}_T S ~=~ T \cap \operatorname{cl}_X S.</math>
{{Collapse top|left=yes|title=Proof}}
Because <math>\operatorname{cl}_X S</math> is a closed subset of <math>X,</math> the intersection <math>T \cap \operatorname{cl}_X S</math> is a closed subset of <math>T</math> (by definition of the [[subspace topology]]), which implies that <math>\operatorname{cl}_T S \subseteq T \cap \operatorname{cl}_X S</math> (because <math>\operatorname{cl}_T S</math> is the {{em|smallest}} closed subset of <math>T</math> containing <math>S</math>). Because <math>\operatorname{cl}_T S</math> is a closed subset of <math>T,</math> from the definition of the subspace topology, there must exist some set <math>C \subseteq X</math> such that <math>C</math> is closed in <math>X</math> and <math>\operatorname{cl}_T S = T \cap C.</math> Because <math>S \subseteq \operatorname{cl}_T S \subseteq C</math> and <math>C</math> is closed in <math>X,</math> the minimality of <math>\operatorname{cl}_X S</math> implies that <math>\operatorname{cl}_X S \subseteq C.</math> Intersecting both sides with <math>T</math> shows that <math>T \cap \operatorname{cl}_X S \subseteq T \cap C = \operatorname{cl}_T S.</math> <math>\blacksquare</math>
{{collapse bottom}}

It follows that <math>S \subseteq T</math> is a dense subset of <math>T</math> [[if and only if]] <math>T</math> is a subset of <math>\operatorname{cl}_X S.</math> 
It is possible for <math>\operatorname{cl}_T S = T \cap \operatorname{cl}_X S</math> to be a proper subset of <math>\operatorname{cl}_X S;</math> for example, take <math>X = \R,</math> <math>S = (0, 1),</math> and <math>T = (0, \infty).</math>

If <math>S, T \subseteq X</math> but <math>S</math> is not necessarily a subset of <math>T</math> then only 
<math display=block>\operatorname{cl}_T (S \cap T) ~\subseteq~ T \cap \operatorname{cl}_X S</math>
is always guaranteed, where this containment could be strict (consider for instance <math>X = \R</math> with the usual topology, <math>T = (-\infty, 0],</math> and <math>S = (0, \infty)</math><ref group=proof>From <math>T := (-\infty, 0]</math> and <math>S := (0, \infty)</math> it follows that <math>S \cap T = \varnothing</math> and <math>\operatorname{cl}_X S = [0, \infty),</math> which implies 
<math display=block>\varnothing ~=~ \operatorname{cl}_T (S \cap T) ~\neq~ T \cap \operatorname{cl}_X S ~=~ \{0\}.</math>
</ref>), although if <math>T</math> happens to an open subset of <math>X</math> then the equality <math>\operatorname{cl}_T (S \cap T) = T \cap \operatorname{cl}_X S</math> will hold (no matter the relationship between <math>S</math> and <math>T</math>). 

{{Collapse top|left=yes|title=Proof}}
Let <math>S, T \subseteq X</math> and assume that <math>T</math> is open in <math>X.</math> Let <math>C := \operatorname{cl}_T (T \cap S),</math> which is equal to <math>T \cap \operatorname{cl}_X (T \cap S)</math> (because <math>T \cap S \subseteq T \subseteq X</math>). The complement <math>T \setminus C</math> is open in <math>T,</math> where <math>T</math> being open in <math>X</math> now implies that <math>T \setminus C</math> is also open in <math>X.</math> Consequently <math>X \setminus (T \setminus C) = (X \setminus T) \cup C</math> is a closed subset of <math>X</math> where <math>(X \setminus T) \cup C</math> contains <math>S</math> as a subset (because if <math>s \in S</math> is in <math>T</math> then <math>s \in T \cap S \subseteq \operatorname{cl}_T (T \cap S) = C</math>), which implies that <math>\operatorname{cl}_X S \subseteq (X \setminus T) \cup C.</math> Intersecting both sides with <math>T</math> proves that <math>T \cap \operatorname{cl}_X S \subseteq T \cap C = C.</math> The reverse inclusion follows from <math>C \subseteq \operatorname{cl}_X (T \cap S) \subseteq \operatorname{cl}_X S.</math> <math>\blacksquare</math>
{{collapse bottom}}

Consequently, if <math>\mathcal{U}</math> is any [[open cover]] of <math>X</math> and if <math>S \subseteq X</math> is any subset then:
<math display=block>\operatorname{cl}_X S = \bigcup_{U \in \mathcal{U}} \operatorname{cl}_U (U \cap S)</math> 
because <math>\operatorname{cl}_U (S \cap U) = U \cap \operatorname{cl}_X S</math> for every <math>U \in \mathcal{U}</math> (where every <math>U \in \mathcal{U}</math> is endowed with the [[subspace topology]] induced on it by <math>X</math>). 
This equality is particularly useful when <math>X</math> is a [[Manifold (mathematics)|manifold]] and the sets in the open cover <math>\mathcal{U}</math> are domains of [[coordinate chart]]s. 
In words, this result shows that the closure in <math>X</math> of any subset <math>S \subseteq X</math> can be computed "locally" in the sets of any open cover of <math>X</math> and then unioned together.
In this way, this result can be viewed as the analogue of the well-known fact that a subset <math>S \subseteq X</math> is closed in <math>X</math> if and only if it is "[[Locally closed set|locally closed]] in <math>X</math>", meaning that if <math>\mathcal{U}</math> is any [[open cover]] of <math>X</math> then <math>S</math> is closed in <math>X</math> if and only if <math>S \cap U</math> is closed in <math>U</math> for every <math>U \in \mathcal{U}.</math>

==Functions and closure==

===Continuity===

{{Main|Continuous function}}

A function <math>f : X \to Y</math> between topological spaces is [[Continuous function|continuous]] if and only if the [[preimage]] of every closed subset of the codomain is closed in the domain; explicitly, this means: <math>f^{-1}(C)</math> is closed in <math>X</math> whenever <math>C</math> is a closed subset of <math>Y.</math> 

In terms of the closure operator, <math>f : X \to Y</math> is continuous if and only if  for every subset <math>A \subseteq X,</math> 
<math display=block>f\left(\operatorname{cl}_X A\right) ~\subseteq~ \operatorname{cl}_Y (f(A)).</math>
That is to say, given any element <math>x \in X</math> that belongs to the closure of a subset <math>A \subseteq X,</math> <math>f(x)</math> necessarily belongs to the closure of <math>f(A)</math> in <math>Y.</math> If we declare that a point <math>x</math> is {{em|close to}} a subset <math>A \subseteq X</math> if <math>x \in \operatorname{cl}_X A,</math> then this terminology allows for a [[plain English]] description of continuity: <math>f</math> is continuous if and only if for every subset <math>A \subseteq X,</math> <math>f</math> maps points that are close to <math>A</math> to points that are close to <math>f(A).</math> 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, <math>f</math> is continuous at a fixed given point <math>x \in X</math> if and only if whenever <math>x</math> is close to a subset <math>A \subseteq X,</math> then <math>f(x)</math> is close to <math>f(A).</math>

===Closed maps===

{{Main|Open and closed maps}}

A function <math>f : X \to Y</math> is a (strongly) [[closed map]] if and only if whenever <math>C</math> is a closed subset of <math>X</math> then <math>f(C)</math> is a closed subset of <math>Y.</math> 
In terms of the closure operator, <math>f : X \to Y</math> is a (strongly) closed map if and only if <math>\operatorname{cl}_Y f(A) \subseteq f\left(\operatorname{cl}_X A\right)</math> for every subset <math>A \subseteq X.</math> 
Equivalently, <math>f : X \to Y</math> is a (strongly) closed map if and only if <math>\operatorname{cl}_Y f(C) \subseteq f(C)</math> for every closed subset <math>C \subseteq X.</math>

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

The [[powerset]] of a set <math>X</math> may be realized as a [[partial order]] [[Category (mathematics)|category]] <math>P</math> in which the objects are subsets and the morphisms are [[inclusion map]]s <math>A \to B</math> whenever <math>A</math> is a subset of <math>B.</math> Furthermore, a topology <math>T</math> on <math>X</math> is a [[subcategory]] of <math>P</math> with inclusion functor <math>I : T \to P.</math> The set of closed subsets containing a fixed subset <math>A \subseteq X</math> can be identified with the [[comma category]] <math>(A \downarrow I).</math> This category — also a partial order — then has initial object <math>\operatorname{cl} A.</math> Thus there is a universal arrow from <math>A</math> to <math>I,</math> given by the inclusion <math>A \to \operatorname{cl} A.</math>

Similarly, since every closed set containing <math>X \setminus A</math> corresponds with an open set contained in <math>A</math> we can interpret the category <math>(I \downarrow X \setminus A)</math> as the set of open subsets contained in <math>A,</math> with [[terminal object]] <math>\operatorname{int}(A),</math> the [[Interior (topology)|interior]] of <math>A.</math>

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 arrow]]s.

==See also==
* {{annotated link|Adherent point}}
* {{annotated link|Closure algebra}}
* [[Closed regular set]], a set equal to the closure of their interior
* {{annotated link|Derived set (mathematics)}}
* {{annotated link|Interior (topology)}}
* {{annotated link|Limit point of a set}}

==Notes==
{{reflist|group=note}}
{{reflist|group=proof}}

==References==
{{reflist}}

==Bibliography==

* {{citation|last=Baker|first=Crump W.|title=Introduction to Topology|year=1991|publisher=Wm. C. Brown Publisher|isbn=0-697-05972-3}}
* {{citation|last=Croom|first=Fred H.|title=Principles of Topology|publisher=Saunders College Publishing|year=1989|isbn=0-03-012813-7|url-access=registration|url=https://archive.org/details/principlesoftopo0000croo}}
* {{citation|last=Gemignani|first=Michael C.|title=Elementary Topology|edition=2nd|year=1990|orig-year=1967|publisher=Dover|isbn=0-486-66522-4}}
* {{citation|last1=Hocking|first1=John G.|first2=Gail S.|last2=Young|title=Topology|year=1988|orig-year=1961|publisher=Dover|isbn=0-486-65676-4|url-access=registration|url=https://archive.org/details/topology00hock_0}}
* {{citation|last=Kuratowski|first=K.|title=Topology|volume=I|publisher=Academic Press|year=1966}}
* {{citation|last=Pervin|first=William J.|title=Foundations of General Topology|year=1965|publisher=Academic Press}}
* {{citation|last=Schubert|first=Horst|title=Topology|year=1968|publisher=Allyn and Bacon}}
* {{Zălinescu Convex Analysis in General Vector Spaces 2002}} <!-- {{sfn|Zălinescu|2002|pp=1-2}} -->

==External links==

* {{springer|title=Closure of a set|id=p/c022630}}

{{Topology|expanded}}

{{DEFAULTSORT:Closure (Topology)}}
[[Category:General topology]]
[[Category:Closure operators]]