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| {{Other uses|Closure (disambiguation)}}
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| 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]].
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| ==Definitions==
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| === Point of closure ===
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| 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).
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| | :<math forcemathmode="mathml">E=mc^2</math> |
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| 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.
| | <!--'''PNG''' (currently default in production) |
| | :<math forcemathmode="png">E=mc^2</math> |
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| 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 (or ''adherent point'') of ''S'' if every neighbourhood of ''x'' contains a point of ''S''.<ref>{{harvnb|Schubert|loc=p. 20}}</ref> Note that this definition does not depend upon whether neighbourhoods are required to be open.
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| | :<math forcemathmode="source">E=mc^2</math> --> |
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| ===Limit point=== | | <span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples]. |
| 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 — 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''.
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| 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.<ref>{{harvnb|Kuratowski|loc=p. 75}}</ref>
| | ==Demos== |
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| 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).
| | Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]: |
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| ===Closure of a set===
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| {{See also|Closure (mathematics)}}
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| The '''closure''' of a set ''S'' is the set of all points of closure of ''S'', that is, the set ''S'' together with all of its limit points.<ref>{{harvnb|Hocking|Young|loc=p. 4}}</ref> 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.<ref>{{harvnb|Croom|loc=p. 104}}</ref>
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| | ** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]]. |
| | ** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]]. |
| | ** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode. |
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| *cl(''S'') is a [[closed set|closed]] superset of ''S''.
| | ==Test pages == |
| *cl(''S'') is the intersection of all [[closed set]]s containing ''S''.
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| *cl(''S'') is the smallest closed set containing ''S''.
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| *cl(''S'') is the union of ''S'' and its [[Boundary (topology)|boundary]] ∂(''S'').
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| *A set ''S'' is closed [[if and only if]] ''S'' = cl(''S'').
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| *If ''S'' is a subset of ''T'', then cl(''S'') is a subset of cl(''T'').
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| *If ''A'' is a closed set, then ''A'' contains ''S'' if and only if ''A'' contains cl(''S'').
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| 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).<ref>{{harvnb|Gemignani|loc=p. 55}}, {{harvnb|Pervin|loc=p. 40}} and {{harvnb|Baker|loc=p. 38}} use the second property as the definition.</ref>
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| 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]]".
| | *[[Inputtypes|Inputtypes (private Wikis only)]] |
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| 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.
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| ==Examples==
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| Consider a sphere in 3 dimensions. 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 be able to distinguish between the interior of 3-ball and the surface, so we distinguish between the open 3-ball, and the closed 3-ball - the closure of the 3-ball. The closure of the open 3-ball is the open 3-ball plus the surface.
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| In [[topological space]]:
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| *In any space, <math>\varnothing=\mathrm{cl}(\varnothing)</math>. | |
| *In any space ''X'', ''X'' = cl(''X'').
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| Giving '''R''' and '''C''' the [[Standard topology|standard (metric) topology]]:
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| *If ''X'' is the Euclidean space '''R''' of [[real number]]s, then cl((0, 1)) = [0, 1].
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| *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'''.
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| *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}.
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| *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]].)
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| On the set of real numbers one can put other topologies rather than the standard one.
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| *If ''X'' = '''R''', where '''R''' has the [[lower limit topology]], then cl((0, 1)) = <nowiki>[</nowiki>0, 1).
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| *If one considers on '''R''' the [[discrete topology]] in which every set is closed (open), then cl((0, 1)) = (0, 1).
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| *If one considers on '''R''' the [[trivial topology]] in which the only closed (open) sets are the empty set and '''R''' itself, then cl((0, 1)) = '''R'''.
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| 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.
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| *In any [[discrete space]], since every set is closed (and also open), every set is equal to its closure.
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| *In any [[indiscrete space]] ''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'', cl(''A'') = ''X''. In other words, every non-empty subset of an indiscrete space is [[Dense set|dense]].
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| 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|relative 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>
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| == Closure operator ==<!-- This section is linked from [[Closure (topology)]] -->
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| {{See also|Closure operator}}
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| A '''closure operator''' on a set ''X'' is a [[map (mathematics)|mapping]] of the [[power set]] of ''X'', <math>\mathcal{P}(X)</math>, into itself which satisfies the [[Kuratowski closure axioms]].
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| Given a [[topological space]] <math>(X, \mathcal{T})</math>, the mapping <sup>−</sup> : ''S'' → ''S''<sup>−</sup> for all {{nowrap|1=''S'' ⊆ ''X''}} is a closure operator on ''X''. Conversely, if '''c''' is a closure operator on a set ''X'', a topological space is obtained by defining the sets ''S'' with '''c'''(''S'') = ''S'' as [[closed set]]s (so their complements are the [[open set]]s of the topology).<ref>{{harvnb|Pervin|loc=p. 41}}</ref>
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| The closure operator <sup>−</sup> is [[Duality (mathematics)|dual]] to the [[Interior (topology)|interior]] operator <sup>o</sup>, in the sense that
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| :''S''<sup>−</sup> = ''X'' \ (''X'' \ ''S'')<sup>o</sup>
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| and also
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| :''S''<sup>o</sup> = ''X'' \ (''X'' \ ''S'')<sup>−</sup>
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| where ''X'' denotes the underlying set of the topological space containing ''S'', and the backslash refers to the [[Complement (set theory)|set-theoretic difference]].
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| 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]].
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| ==Facts about closures==
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| The set <math>S</math> is [[closed set|closed]] if and only if <math>Cl(S)=S</math>. In particular:
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| * The closure of the [[empty set]] is the empty set;
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| * The closure of <math>X</math> itself is <math>X</math>.
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| * 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.
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| * 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.
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| * 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.
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| 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>.
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| ==Categorical interpretation==
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| One may elegantly define the closure operator in terms of universal arrows, as follows.
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| 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 — also a partial order — 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>.
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| 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''.
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| 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.
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| ==See also==
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| * [[Closure algebra]]
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| ==Notes==
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| {{reflist|3}}
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| ==References==
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| * {{citation|first=Crump W.|last=Baker|title=Introduction to Topology|year=1991|publisher=Wm. C. Brown Publisher|isbn=0-697-05972-3}}
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| * {{citation|first=Fred H.|last=Croom|title=Principles of Topology|publisher=Saunders College Publishing|year=1989|isbn=0-03-012813-7}}
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| * {{citation|first=Michael C.|last=Gemignani|title=Elementary Topology|edition=2nd|year=1990|origyear=1967|publisher=Dover|isbn=0-486-66522-4}}
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| * {{citation|first1=John G.|last1=Hocking|first2=Gail S.|last2=Young|title=Topology|year=1988|origyear=1961|publisher=Dover|isbn=0-486-65676-4}}
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| * {{citation|first=K.|last=Kuratowski|title=Topology|volume=I|publisher=Academic Press|year=1966}}
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| * {{citation|first=William J.|last=Pervin|title=Foundations of General Topology|year=1965|publisher=Academic Press}}
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| * {{citation|first=Horst|last=Schubert|title=Topology|year=1968|publisher=Allyn and Bacon}}
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| ==External links==
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| * {{springer|title=Closure of a set|id=p/c022630}}
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| {{DEFAULTSORT:Closure (Topology)}}
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| [[Category:General topology]]
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| [[Category:Closure operators]]
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