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{{Unreferenced|date=March 2007}}
In [[mathematics]], the '''closure''' of a subset ''S'' in a [[topological space]] consists of all [[Topology glossary#P|point]]s in ''S'' plus the [[limit points]] of ''S''. The closure of ''S'' is also defined as the union of ''S'' and its [[Boundary_(topology)|boundary]]. Intuitively, these are all the points in ''S'' and "near" ''S''. A point which is in the closure of ''S'' is a [[adherent point|point of closure]] of ''S''. The notion of closure is in many ways [[duality (mathematics)|dual]] to the notion of [[interior (topology)|interior]].
 
==Definitions==
 
=== Point of closure ===
 
For ''S'' a subset of a [[Euclidean space]], ''x'' is a point of closure of ''S'' if every [[open ball]] centered at ''x'' contains a point of ''S'' (this point may be ''x'' itself).
 
This definition generalises to any subset ''S'' of a [[metric space]] ''X''. Fully expressed, for ''X'' a metric space with metric ''d'', ''x'' is a point of closure of ''S'' if for every ''r'' > 0, there is a ''y'' in ''S'' such that the distance ''d''(''x'', ''y'') < ''r''. (Again, we may have ''x'' = ''y''.) Another way to express this is to say that ''x'' is a point of closure of ''S'' if the distance ''d''(''x'', ''S'') := [[infimum|inf]]{''d''(''x'', ''s'') : ''s'' in ''S''} = 0.
 
This definition generalises to [[topological space]]s by replacing "open ball" or "ball" with "[[Topology glossary#N|neighbourhood]]". Let ''S'' be a subset of a topological space ''X''. Then ''x'' is a point of closure of ''S'' if every neighbourhood of ''x'' contains a point of ''S''. Note that this definition does not depend upon whether neighbourhoods are required to be open.
 
===Limit point===
The definition of a point of closure is closely related to the definition of a [[limit point]]. The difference between the two definitions is subtle but important &mdash; namely, in the definition of limit point, every neighborhood of the point ''x'' in question must contain a point of the set ''other than x itself''.
 
Thus, every limit point is a point of closure, but not every point of closure is a limit point. A point of closure which is not a limit point is an [[isolated point]]. In other words, a point ''x'' is an isolated point of ''S'' if it is an element of ''S'' and if there is a neighbourhood of ''x'' which contains no other points of ''S'' other than ''x'' itself.
 
For a given set ''S'' and point ''x'', ''x'' is a point of closure of ''S'' if and only if ''x'' is an element of ''S'' or ''x'' is a limit point of ''S'' (or both).
 
===Closure of a set===
{{See also|Closure (mathematics)}}
 
The '''closure''' of a set ''S'' is the set of all points of closure of ''S''. The closure of ''S'' is denoted cl(''S''), Cl(''S''), <math>\scriptstyle\bar{S}</math> or <math>\scriptstyle S^-</math>. The closure of a set has the following properties.
 
*cl(''S'') is a [[closed set|closed]] superset of ''S''.
*cl(''S'') is the intersection of all [[closed set]]s containing ''S''.
*cl(''S'') is the smallest closed set containing ''S''.
*A set ''S'' is closed [[if and only if]] ''S'' = cl(''S'').
*If ''S'' is a subset of ''T'', then cl(''S'') is a subset of cl(''T'').
*If ''A'' is a closed set, then ''A'' contains ''S'' if and only if ''A'' contains cl(''S'').
 
Sometimes the second or third property above is taken as the ''definition'' 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]]), cl(''S'') is the set of all [[limit of a sequence|limits]] of all convergent [[sequence]]s of points in ''S''. For a general topological space, this statement remains true if one replaces "sequence" by "[[net (mathematics)|net]]" or "[[filter (mathematics)|filter]]".
 
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 (topology)#Closure operator|closure operator]] below.
 
==Examples==
* In any space, <math>\varnothing=\mathrm{cl}(\varnothing)</math>.
*In any space ''X'', ''X'' = cl(''X'').
Giving '''R''' and '''C''' the [[Standard topology|standard (metric) topology]]:
*If ''X'' is the Euclidean space '''R''' of [[real number]]s, then cl((0, 1)) = [0, 1].
*If ''X'' is the Euclidean space '''R''', then the closure of the set '''Q''' of [[rational number]]s is the whole space '''R'''. We say that '''Q''' is [[dense (topology)|dense]] in '''R'''.
*If ''X'' is the [[complex number|complex plane]] '''C''' = '''R'''<sup>2</sup>, then cl({''z'' in '''C''' : |''z''| > 1}) = {''z'' in '''C''' : |''z''| ≥ 1}.
*If ''S'' is a [[finite set|finite]] subset of a Euclidean space, then cl(''S'') = ''S''. (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 ''X'' = '''R''', where '''R''' has the [[lower limit topology]], then cl((0, 1)) = <nowiki>[</nowiki>0, 1).
*If one considers on '''R''' the [[discrete topology]] in which every set is open (closed), then cl((0, 1)) = (0, 1).
*If one considers on '''R''' the [[trivial topology]] in which the only open (closed) sets are the empty set and '''R''' itself, then cl((0, 1)) = '''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 any [[discrete space]], since every set is open (closed), every set is equal to its closure.
*In any [[indiscrete space]] ''X'', since the only open (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'', cl(''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 [[subspace topology]] induced by the Euclidean space '''R''', and if ''S'' = {''q'' in '''Q''' : ''q''<sup>2</sup> > 2, ''q'' > 0}, then ''S'' is closed in '''Q''', and the closure of ''S'' in '''Q''' is ''S''; however, the closure of ''S'' in the Euclidean space '''R''' is the set of all ''real numbers'' greater than ''or equal to'' <math>\sqrt2.</math>
 
== Closure operator ==<!-- This section is linked from [[Closure (topology)]] -->
{{See also|Closure operator}}
 
The '''closure operator''' <sup>&minus;</sup> is [[Duality (mathematics)|dual]] to the  [[Interior (topology)|interior]] operator <sup>o</sup>, in the sense that
 
:''S''<sup>&minus;</sup> = ''X'' \ (''X'' \ ''S'')<sup>o</sup>
 
and also
 
:''S''<sup>o</sup> = ''X'' \ (''X'' \ ''S'')<sup>&minus;</sup>
 
where ''X'' denotes the [[topological space]] containing ''S'', and the backslash refers to the [[Complement (set theory)|set-theoretic difference]].
 
Therefore, the abstract theory of closure operators and the [[Kuratowski closure axioms]] can be easily translated into the language of interior operators, by replacing sets with their [[Complement (set theory)|complements]].
 
==Facts about closures==
The set <math>S</math> is [[closed set|closed]] if and only if <math>Cl(S)=S</math>. In particular, the closure of the [[empty set]] is the empty set, and 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.
 
If <math>A</math> is a [[topological subspace|subspace]] of <math>X</math> containing <math>S</math>, then the closure of <math>S</math> computed in <math>A</math> is equal to the intersection of <math>A</math> and the closure of <math>S</math> computed in <math>X</math>: <math>Cl_A(S) = A\cap Cl_X(S)</math>. In particular, <math>S</math> is dense in <math>A</math> [[if and only if]] <math>A</math> is a subset of <math>Cl_X(S)</math>.
 
==Categorical interpretation==
One may elegantly define the closure operator in terms of universal arrows, as follows.
 
The [[powerset]] of a set ''X'' may be realized as a [[partial order]] [[category (mathematics)|category]] ''P'' in which the objects are subsets and the morphisms are inclusions <math>A \to B</math> whenever ''A'' is a subset of ''B''. Furthermore, a topology ''T'' on ''X'' is a subcategory of ''P'' 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 &mdash; also a partial order &mdash; then has initial object Cl(''A''). Thus there is a universal arrow from ''A'' to ''I'', given by the inclusion <math>A \to Cl(A)</math>.
 
Similarly, since every closed set containing ''X'' \ ''A'' corresponds with an open set contained in ''A'' we can interpret the category <math> (I \downarrow X \setminus A)</math> as the set of open subsets contained in ''A'', with terminal object <math>int(A)</math>, the [[interior (topology)|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|algebraic]]), since all are examples of universal arrows.
 
==See also==
* [[Closure algebra]]
 
==External links==
* {{springer|title=Closure of a set|id=p/c022630}}
 
{{DEFAULTSORT:Closure (Topology)}}
[[Category:General topology]]
[[Category:Closure operators]]

Revision as of 06:35, 3 March 2014

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