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[[Image:Interior illustration.svg|right|thumb|The point ''x'' is an interior point of ''S'', since  it is contained within ''S'' together with an open ball around it. The point ''y'' is on the boundary of ''S''.]]
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In [[mathematics]], specifically in [[general topology|topology]], the '''interior''' of a set ''S'' of points of a [[topological space]] consists of all [[Topology glossary#P|point]]s of ''S'' that do not belong to the [[Boundary_(topology)|boundary]] of ''S''. A point that is in the interior of ''S'' is an '''interior point''' of ''S''.
 
Equivalently the interior of ''S'' is the [[Absolute complement|complement]] of the [[closure (topology)|closure]] of the complement of ''S''. In this sense interior and closure are [[Duality_(mathematics)#Duality_in_logic_and_set_theory|dual]] notions.  
 
The '''exterior''' of a set is the interior of its complement, equivalently the complement of its closure; it consists of the points that are in neither the set nor its boundary. The interior, boundary, and exterior of a subset together [[partition of a set|partition]] the whole space into three blocks (or fewer when one or more of these is empty). The interior and exterior are always [[open set|open]] while the boundary is always [[closed set|closed]].  Sets with empty interior have been called '''boundary sets'''.<ref>{{Cite journal
  | last = Kuratowski
  | first = Kazimierz
  | authorlink = Kazimierz Kuratowski
  | title = Sur l'Operation Ā de l'Analysis Situs
  | url = http://matwbn.icm.edu.pl/ksiazki/fm/fm3/fm3121.pdf
  | journal = Fundamenta Mathematicae
  | volume = 3
  | pages = 182–199
  | publisher = Polish Academy of Sciences
  | location = Warsaw
  | year = 1922
  | issn = 0016-2736}}</ref>
 
== Definitions ==
=== Interior point ===
If ''S'' is a subset of a [[Euclidean space]], then ''x'' is an interior point of ''S'' if there exists an [[open set]] centered at ''x'' which is contained in ''S''.
 
This definition generalizes to any subset ''S'' of a [[metric space]] ''X''. Fully expressed, if ''X'' is a metric space with metric ''d'', then ''x'' is an interior point of ''S'' if there exists ''r'' > 0, such that ''y'' is in ''S'' whenever the distance ''d''(''x'', ''y'') < ''r''.
 
This definition generalises to [[topological space]]s by replacing "open ball" with "[[Neighbourhood (mathematics)|neighbourhood]]". Let ''S'' be a subset of a topological space ''X''. Then ''x'' is an interior point of ''S'' if there exists a neighbourhood of ''x'' which is contained in ''S''. Note that this definition does not depend upon whether neighbourhoods are required to be open. If neighbourhoods are not required to be open then ''S'' will automatically be a neighbourhood of ''x'' if ''S'' contains a neighbourhood of ''x''.
 
=== Interior of a set ===
 
The '''interior''' of a set ''S'' is the set of all interior points of ''S''. The interior of ''S'' is denoted int(''S''), Int(''S''), or ''S''<sup>o</sup>. The interior of a set has the following properties.
 
*int(''S'') is an [[Open set|open]] subset of ''S''.
*int(''S'') is the union of all open sets contained in ''S''.
*int(''S'') is the largest open set contained in ''S''.
*A set ''S'' is open [[if and only if]] ''S'' = int(''S'').
*int(int(''S'')) = int(''S'') ([[idempotent|idempotence]]).
*If ''S'' is a subset of ''T'', then int(''S'') is a subset of int(''T'').
*If ''A'' is an open set, then ''A'' is a subset of ''S'' if and only if ''A'' is a subset of int(''S'').
 
Sometimes the second or third property above is taken as the ''definition'' of the topological interior.
 
Note that these properties are also satisfied if "interior", "subset", "union", "contained in", "largest" and "open" are replaced by "closure", "superset", "intersection", "which contains", "smallest", and "closed", respectively. For more on this matter, see [[Interior (topology)#Interior operator|interior operator]] below.
 
== Examples ==
 
*In any space, the interior of the empty set is the empty set.
*In any space ''X'', if <math> A\subset X </math>, int(''A'') is contained in ''A''.
*If ''X'' is the Euclidean space <math>\mathbb{R}</math> of [[real number]]s, then int([0, 1]) = (0, 1).
*If ''X'' is the Euclidean space <math>\mathbb{R}</math>, then the interior of the set <math>\mathbb{Q}</math> of [[rational number]]s is empty.
*If ''X'' is the [[complex number|complex plane]] <math>\mathbb{C} = \mathbb{R}^2</math>, then int<math>(\{z\in \mathbb{C} : |z| \geq 1\}) = \{z\in \mathbb{C} : |z| > 1\}.</math>
*In any Euclidean space, the interior of any [[finite set|finite]] set is the empty set.
 
On the set of real numbers one can put other topologies rather than the standard one.
 
*If <math> X = \mathbb{R}</math>, where <math>\mathbb{R}</math> has the [[lower limit topology]], then int([0, 1]) = <nowiki>[0, 1)</nowiki>.  
*If one considers on <math>\mathbb{R}</math> the topology in which every set is open, then int([0, 1]) = [0, 1].
*If one considers on <math>\mathbb{R}</math> the topology in which the only open sets are the empty set and <math>\mathbb{R}</math> itself, then int([0, 1]) is the empty set.  
 
These examples show that the interior 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, every set is equal to its interior.
*In any [[indiscrete space]] ''X'', since the only open sets are the empty set and ''X'' itself, we have int(''X'') = ''X'' and for every [[subset|proper subset]] ''A'' of ''X'', int(''A'') is the empty set.
 
== Interior operator ==<!-- This section is linked from above -->
 
The '''interior operator''' <sup>o</sup> is dual to the [[Closure (topology)|closure]] operator <sup>—</sup>, in the sense that
 
:''S''<sup>o</sup> = ''X'' \ (''X'' \ ''S'')<sup>—</sup>,
 
and also
 
:''S''<sup>—</sup> = ''X'' \ (''X'' \ ''S'')<sup>o</sup>
 
where ''X'' is 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 complements.  
 
== Exterior of a set ==
{{main|Exterior (topology)}}
The '''exterior''' of a subset ''S'' of a topological space ''X'', denoted ext(''S'') or Ext(''S''), is the interior int(''X''&nbsp;\&nbsp;''S'') of its relative complement. Alternatively, it can be defined as ''X''&nbsp;\&nbsp;''S''<sup>—</sup>, the complement of the closure of ''S''. Many properties follow in a straightforward way from those of the interior operator, such as the following.
 
*ext(''S'') is an open set that is disjoint with ''S''.
*ext(''S'') is the union of all open sets that are disjoint with ''S''.
*ext(''S'') is the largest open set that is disjoint with ''S''.
*If ''S'' is a subset of ''T'', then ext(''S'') is a superset of ext(''T'').
 
Unlike the interior operator, ext is not idempotent, but the following holds:
 
*ext(ext(''S'')) is a superset of int(''S'').
 
==See also==
* [[Algebraic interior]]
* [[Interior algebra]]
* [[Jordan curve theorem]]
* [[Quasi-relative interior]]
* [[Relative interior]]
 
==References==
{{Reflist}}
 
==External links==
*{{PlanetMath|id=3123|title=Interior}}
 
{{Functional Analysis}}
 
[[Category:General topology]]
[[Category:Closure operators]]

Latest revision as of 13:56, 17 November 2014

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An air rim clamp tire changer is basically the identical because the high volume capacity mannequin, besides that it is meant for smaller in dimension automobiles. However, it was specifically designed to deal with powerful to vary tires, equivalent to low profiles and run flats. This aspect is helpful to any operator as a result of, as any semi- skilled would admit, you do run into most of these tires as a rule and it is good to be prepared by having a software such as this to help you. It gives you the assistance that it's essential full any kind of job precisely.

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