View source for Interior (topology)

{{Short description|Largest open subset of some given set}}
[[Image:Interior illustration.svg|right|thumb|The point {{mvar|x}} is an interior point of {{mvar|S}}.
The point {{mvar|y}} is on the boundary of {{mvar|S}}.]]
In [[mathematics]], specifically in [[general topology|topology]],
the '''interior''' of a [[subset]] {{mvar|S}} of a [[topological space]] {{mvar|X}} is the [[Union (set theory)|union]] of all subsets of {{mvar|S}} that are [[Open set|open]] in {{mvar|X}}.
A point that is in the interior of {{mvar|S}} is an '''interior point''' of {{mvar|S}}.

The interior of {{mvar|S}} is the [[Absolute complement|complement]] of the [[Closure (topology)|closure]] of the complement of {{mvar|S}}.
In this sense interior and closure are [[Duality_(mathematics)#Duality_in_logic_and_set_theory|dual]] notions.

The '''exterior''' of a set {{mvar|S}} is the complement of the closure of {{mvar|S}}; it consists of the points that are in neither the set nor its [[Boundary (topology)|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 set|empty]]).

The interior and exterior of a [[closed curve]] are a slightly different concept; see the [[Jordan curve theorem]].

==Definitions==

===Interior point===

If <math>S</math> is a subset of a [[Euclidean space]], then <math>x</math> is an interior point of <math>S</math> if there exists an [[open ball]] centered at <math>x</math> which is completely contained in <math>S.</math>
(This is illustrated in the introductory section to this article.)

This definition generalizes to any subset <math>S</math> of a [[metric space]] <math>X</math> with metric <math>d</math>: <math>x</math> is an interior point of <math>S</math> if there exists a real  number <math>r > 0,</math> such that <math>y</math> is in <math>S</math> whenever the distance <math>d(x, y) < r.</math>

This definition generalizes to [[topological space]]s by replacing "open ball" with "[[open set]]".
If <math>S</math> is a subset of a topological space <math>X</math> then <math>x</math> is an {{em|interior point}} of <math>S</math> in <math>X</math> if <math>x</math> is contained in an open subset of <math>X</math> that is completely contained in <math>S.</math>
(Equivalently, <math>x</math> is an interior point of <math>S</math> if <math>S</math> is a [[Neighbourhood (mathematics)|neighbourhood]] of <math>x.</math>)

===Interior of a set===

The '''interior''' of a subset <math>S</math> of a topological space <math>X,</math> denoted by <math>\operatorname{int}_X S</math> or <math>\operatorname{int} S</math> or <math>S^\circ,</math> can be defined in any of the following equivalent ways:
# <math>\operatorname{int} S</math> is the largest open subset of <math>X</math> contained in <math>S.</math>
# <math>\operatorname{int} S</math> is the union of all open sets of <math>X</math> contained in <math>S.</math>
# <math>\operatorname{int} S</math> is the set of all interior points of <math>S.</math>
If the space <math>X</math> is understood from context then the shorter notation <math>\operatorname{int} S</math> is usually preferred to <math>\operatorname{int}_X S.</math>

==Examples==
[[File:Set of real numbers with epsilon-neighbourhood.svg|thumb|<math>a</math> is an interior point of <math>M</math> because there is an ε-neighbourhood of a which is a subset of <math>M.</math>]]

*In any space, the interior of the [[empty set]] is the empty set.
*In any space <math>X,</math> if <math>S \subseteq X,</math> then <math>\operatorname{int} S \subseteq S.</math>
*If <math>X</math> is the [[real line]] <math>\Reals</math> (with the standard topology), then <math>\operatorname{int} ([0, 1]) = (0, 1)</math> whereas the interior of the set <math>\Q</math> of [[rational number]]s is empty: <math>\operatorname{int} \Q = \varnothing.</math>
*If <math>X</math> is the [[Complex number|complex plane]] <math>\Complex,</math> then <math>\operatorname{int} (\{z \in \Complex : |z| \leq 1\}) = \{z \in \Complex : |z| < 1\}.</math>
*In any [[Euclidean space]], the interior of any [[finite set]] is the empty set.

On the set of [[real number]]s, one can put other topologies rather than the standard one:

*If <math>X</math> is the real numbers <math>\Reals</math> with the [[lower limit topology]], then <math>\operatorname{int} ([0, 1]) = [0, 1).</math>
*If one considers on <math>\Reals</math> the topology in which [[Discrete topology|every set is open]], then <math>\operatorname{int} ([0, 1]) = [0, 1].</math>
*If one considers on <math>\Reals</math> the topology in which the only open sets are the empty set and <math>\Reals</math> itself, then <math>\operatorname{int} ([0, 1])</math> 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]] <math>X,</math> since the only open sets are the empty set and <math>X</math> itself, <math>\operatorname{int} X = X</math> and for every [[subset|proper subset]] <math>S</math> of <math>X,</math> <math>\operatorname{int} S</math> is the empty set.

==Properties==

Let <math>X</math> be a topological space and let <math>S</math> and <math>T</math> be subsets of <math>X.</math>

* <math>\operatorname{int} S</math> is [[Open set|open]] in <math>X.</math>
* If <math>T</math> is open in <math>X</math> then <math>T \subseteq S</math> if and only if <math>T \subseteq \operatorname{int} S.</math>
* <math>\operatorname{int} S</math> is an open subset of <math>S</math> when <math>S</math> is given the [[subspace topology]].
* <math>S</math> is an open subset of <math>X</math> [[if and only if]] <math>\operatorname{int} S = S.</math>
* {{em|Intensive}}: <math>\operatorname{int} S \subseteq S.</math>
* [[Idempotent|{{em|Idempotence}}]]: <math>\operatorname{int} (\operatorname{int} S) = \operatorname{int} S.</math>
* {{em|Preserves}}/{{em|[[Distributive property|distributes over]] binary intersection}}: <math>\operatorname{int} (S \cap T) = (\operatorname{int} S) \cap (\operatorname{int} T).</math>
** However, the interior operator does not distribute over unions since only <math>\operatorname{int} (S \cup T) ~\supseteq~ (\operatorname{int} S) \cup (\operatorname{int} T)</math> is guaranteed in general and equality might not hold.<ref group="note" name="mnemonicInteriorAndIntersection" /> For example, if <math>X = \Reals, S = (-\infty, 0],</math> and <math>T = (0, \infty)</math> then <math>(\operatorname{int} S) \cup (\operatorname{int} T) = (-\infty, 0) \cup (0, \infty) = \Reals \setminus \{0\}</math> is a proper subset of <math>\operatorname{int} (S \cup T) = \operatorname{int} \Reals = \Reals.</math>
* {{em|Monotone}}/{{em|nondecreasing with respect to <math>\subseteq</math>}}: If <math>S \subseteq T</math> then <math>\operatorname{int} S \subseteq \operatorname{int} T.</math>

Other properties include:

* If <math>S</math> is closed in <math>X</math> and <math>\operatorname{int} T = \varnothing</math> then <math>\operatorname{int} (S \cup T) = \operatorname{int} S.</math>

'''Relationship with closure'''

The above statements will remain true if all instances of the symbols/words
:"interior", "int", "open", "subset", and "largest"
are respectively replaced by
:"[[Closure (topology)|closure]]", "cl", "closed", "superset", and "smallest"
and the following symbols are swapped:
# "<math>\subseteq</math>" swapped with "<math>\supseteq</math>"
# "<math>\cup</math>" swapped with "<math>\cap</math>"
For more details on this matter, see [[Interior (topology)#Interior operator|interior operator]] below or the article [[Kuratowski closure axioms]].

==Interior operator==
<!-- This section is linked from above -->

The '''interior operator''' <math>\operatorname{int}_X</math> is dual to the [[Closure (topology)|closure]] operator, which is denoted by <math>\operatorname{cl}_X</math> or by an overline <sup>—</sup>, in the sense that
<math display="block">\operatorname{int}_X S = X \setminus \overline{(X \setminus S)}</math>
and also
<math display="block">\overline{S} = X \setminus \operatorname{int}_X (X \setminus S),</math>
where <math>X</math> is the [[topological space]] containing <math>S,</math> and the backslash <math>\,\setminus\,</math> denotes [[Complement (set theory)|set-theoretic difference]].
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 complements in <math>X.</math>

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

{{Math theorem|name=Theorem<ref name="Zalinescu 2002 p. 33">{{cite book|last=Zalinescu|first=C|title=Convex analysis in general vector spaces|publisher=World Scientific|publication-place=River Edge, N.J. London|year=2002|isbn=981-238-067-1| oclc=285163112| page=33}}</ref> |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} \biggl( \bigcup_{i \in \N} \operatorname{int}_X S_i \biggr)
= {\operatorname{cl}_X \operatorname{int}_X} \biggl( \bigcup_{i \in \N} S_i \biggr).
</math>
*If each <math>S_i</math> is open in <math>X</math> then <math display="block">
{\operatorname{int}_X} \biggl( \bigcap_{i \in \N} \operatorname{cl}_X S_i \biggr)
= {\operatorname{int}_X \operatorname{cl}_X} \biggl( \bigcap_{i \in \N} S_i \biggr).
</math>
}}

The result above implies that every complete metric space is a [[Baire space]].

==Exterior of a set==

The '''exterior''' of a subset <math>S</math> of a topological space <math>X,</math> denoted by <math>\operatorname{ext}_X S</math> or simply <math>\operatorname{ext} S,</math> is the largest open set [[disjoint (sets)|disjoint]] from <math>S,</math> namely, it is the union of all open sets in <math>X</math> that are disjoint from <math>S.</math>  The exterior is the interior of the complement, which is the same as the complement of the closure;{{sfn|Bourbaki|1989|p=24}} in formulas,
<math display="block">\operatorname{ext} S = \operatorname{int}(X \setminus S) = X \setminus \overline{S}.</math>

Similarly, the interior is the exterior of the complement:
<math display="block">\operatorname{int} S = \operatorname{ext}(X \setminus S).</math>

The interior, [[Boundary (topology)|boundary]], and exterior of a set <math>S</math> together [[partition of a set|partition]] the whole space into three blocks (or fewer when one or more of these is empty):
<math display="block">X = \operatorname{int} S \cup \partial S \cup \operatorname{ext} S,</math>
where <math>\partial S</math> denotes the boundary of <math>S.</math>{{sfn|Bourbaki|1989|p=25}}  The interior and exterior are always [[open set|open]], while the boundary is [[Closed set|closed]].

Some of the properties of the exterior operator are unlike those of the interior operator:
* The exterior operator reverses inclusions; if <math>S \subseteq T,</math> then <math>\operatorname{ext} T \subseteq \operatorname{ext} S.</math>
* The exterior operator is not [[idempotent]].  It does have the property that <math>\operatorname{int} S \subseteq \operatorname{ext}\left(\operatorname{ext} S\right).</math>

==Interior-disjoint shapes==

[[Image:Interior-disjoint.svg|right|thumb|The red shapes are not interior-disjoint with the blue Triangle. The green and the yellow shapes are interior-disjoint with the blue Triangle, but only the yellow shape is entirely disjoint from the blue Triangle.]]

Two shapes <math>a</math> and <math>b</math> are called ''interior-disjoint'' if the intersection of their interiors is empty.
Interior-disjoint shapes may or may not intersect in their boundary.

==See also==

* {{annotated link|Algebraic interior}}
* {{annotated link|DE-9IM}}
* {{annotated link|Interior algebra}}
* {{annotated link|Jordan curve theorem}}
* {{annotated link|Quasi-relative interior}}
* {{annotated link|Relative interior}}

==References==

{{reflist}}

{{reflist|group=note|refs=
<ref name="mnemonicInteriorAndIntersection">The analogous identity for the [[Closure (topology)|closure operator]] is <math>\operatorname{cl} (S \cup T) = (\operatorname{cl} S) \cup (\operatorname{cl} T).</math> These identities may be remembered with the following mnemonic. Just as the intersection <math>\cap</math> of two open sets is open, so too does the interior operator distribute over intersections <math>\cap;</math> explicitly: <math>\operatorname{int} (S \cap T) = (\operatorname{int} S) \cap (\operatorname{int} T).</math> And similarly, just as the union <math>\cup</math> of two closed sets is closed, so too does the [[Closure (topology)|closure operator]] distribute over unions <math>\cup;</math> explicitly: <math>\operatorname{cl} (S \cup T) = (\operatorname{cl} S) \cup (\operatorname{cl} T).</math></ref>
}}

==Bibliography==

* {{Bourbaki General Topology Part I Chapters 1-4}} <!-- {{sfn | Bourbaki | 1989 | p=}} -->
* {{Dixmier General Topology}} <!-- {{sfn | Dixmier | 1984 | p=}} -->
* {{Császár General Topology}} <!-- {{sfn | Császár | 1978 | p=}} -->
* {{Dugundji Topology}} <!-- {{sfn | Dugundji | 1966 | p=}} -->
* {{Joshi Introduction to General Topology}} <!-- {{sfn | Joshi | 1983 | p=}} -->
* {{Kelley General Topology}} <!-- {{sfn | Kelley | 1975 | p=}} -->
* {{Munkres Topology|edition=2}} <!-- {{sfn | Munkres | 2000 | p=}} -->
* {{Schubert Topology}} <!-- {{sfn | Schubert | 1968 | p=}} -->
* {{Wilansky Topology for Analysis 2008}} <!-- {{sfn|Wilansky|2008|p=}} -->
* {{Willard General Topology}} <!-- {{sfn|Willard|2004|p=}} -->

==External links==

* {{PlanetMath|id=3123|title=Interior}}

{{Topology|expanded}}

[[Category:Closure operators]]
[[Category:General topology]]