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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformations, such ...
, the interior of a
subset In mathematics, Set (mathematics), set ''A'' is a subset of a set ''B'' if all Element (mathematics), elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are ...
of a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
is the
union Union commonly refers to: * Trade union, an organization of workers * Union (set theory), in mathematics, a fundamental operation on sets Union may also refer to: Arts and entertainment Music * Union (band), an American rock group ** ''Un ...
of all subsets of that are
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...
in . A point that is in the interior of is an interior point of . The interior of is the
complement A complement is something that completes something else. Complement may refer specifically to: The arts * Complement (music), an interval that, when added to another, spans an octave ** Aggregate complementation, the separation of pitch-class ...
of the closure of the complement of . In this sense interior and closure are
dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...
notions. The exterior of a set is the complement of the closure of ; it consists of the points that are in neither the set nor its
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
. The interior, boundary, and exterior of a subset together
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
the whole space into three blocks (or fewer when one or more of these is
empty Empty may refer to: ‍ Music Albums * ''Empty'' (God Lives Underwater album) or the title song, 1995 * ''Empty'' (Nils Frahm album), 2020 * ''Empty'' (Tait album) or the title song, 2001 Songs * "Empty" (The Click Five song), 2007 * ...
).


Definitions


Interior point

If is a subset of a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, then is an interior point of if there exists an
open ball In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...
centered at which is completely contained in . (This is illustrated in the introductory section to this article.) This definition generalizes to any subset of a
metric space In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
with metric : is an interior point of if there exists r > 0, such that is in whenever the distance d(x, y) < r. This definition generalises to
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s by replacing "open ball" with "
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are suf ...
". Let be a subset of a topological space . Then is an interior point of if is contained in an open subset of which is completely contained in . (Equivalently, is an interior point of if is a
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
of .)


Interior of a set

The interior of a subset of a topological space , denoted by \operatornameS or \operatornameS or S^\circ, can be defined in any of the following equivalent ways: # is the largest open subset of contained (as a subset) in # is the union of all open sets of contained in # is the set of all interior points of


Examples

*In any space, the interior of the empty set is the empty set. *In any space , if S \subseteq X, then \operatorname S \subseteq S. *If is the
real line In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
\Reals (with the standard topology), then . *If is the real line \Reals, then the interior of the set \Q of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ration ...
s is empty. *If is the
complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by the ...
\Complex, then \operatorname(\) = \. *In any Euclidean space, the interior of any
finite set In mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, :\ is a finite set with five elements. Th ...
is the empty set. On the set of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
s, one can put other topologies rather than the standard one: *If is the real numbers \Reals with the
lower limit topology In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set \mathbb of real numbers; it is different from the standard topology on \mathbb (generated by the open intervals) and has a number of in ...
, then . *If one considers on \Reals the topology in which every set is open, then . *If one considers on \Reals the topology in which the only open sets are the empty set and \Reals itself, then 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 In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
, since every set is open, every set is equal to its interior. *In any
indiscrete space In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
, since the only open sets are the empty set and itself, \operatorname X = X and for every
proper subset In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of , \operatorname S is the empty set.


Properties

Let be a topological space and let and be subsets of . * \operatorname S is
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...
in . * If is open in then T \subseteq S if and only if T \subseteq \operatorname S. * \operatorname S is an open subset of when is given the
subspace topology In topology and related areas of mathematics, a subspace of a topological space ''X'' is a subset ''S'' of ''X'' which is equipped with a topology induced from that of ''X'' called the subspace topology (or the relative topology, or the induced to ...
. * is an open subset of
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bicondi ...
\operatorname S = S. * : \operatorname S \subseteq S. * : \operatorname (\operatorname S) = \operatorname S. * /: \operatorname (S \cap T) = (\operatorname S) \cap (\operatorname T). ** However, the interior operator does not distribute over unions since only \operatorname (S \cup T) ~\supseteq~ (\operatorname S) \cup (\operatorname T) is guaranteed in general and equality might not hold. For example, if X = \Reals, S = (-\infty, 0], and T = (0, \infty) then (\operatorname S) \cup (\operatorname T) = (-\infty, 0) \cup (0, \infty) = \Reals \setminus \ is a proper subset of \operatorname (S \cup T) = \operatorname \Reals = \Reals. * /: If S \subseteq T then \operatorname S \subseteq \operatorname T. Other properties include: * If is closed in and \operatorname T = \varnothing then \operatorname (S \cup T) = \operatorname S. 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: # "\subseteq" swapped with "\supseteq" # "\cup" swapped with "\cap" For more details on this matter, see
interior operator In mathematics, a closure operator on a set ''S'' is a function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X,Y\subseteq S : Closure operators are det ...
below or the article
Kuratowski closure axioms In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formal ...
.


Interior operator

The interior operator \operatorname_X is dual to the Closure (topology), closure operator, which is denoted by \operatorname_X or by an overline , in the sense that \operatorname_X S = X \setminus \overline and also \overline = X \setminus \operatorname_X (X \setminus S), where X is the
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
containing S, and the backslash \,\setminus\, denotes
set-theoretic difference In set theory, the complement of a set , often denoted by (or ), is the set of elements not in . When all sets in the universe, i.e. all sets under consideration, are considered to be members of a given set , the absolute complement of is the ...
. Therefore, the abstract theory of closure operators and the
Kuratowski closure axioms In topology and related branches of mathematics, the Kuratowski closure axioms are a set of axioms that can be used to define a topological structure on a set. They are equivalent to the more commonly used open set definition. They were first formal ...
can be readily translated into the language of interior operators, by replacing sets with their complements in X. In general, the interior operator does not commute with unions. However, in a
complete metric space In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the boun ...
the following result does hold: The result above implies that every complete metric space is a
Baire space In mathematics, a topological space X is said to be a Baire space if countable unions of closed sets with empty interior also have empty interior. According to the Baire category theorem, compact Hausdorff spaces and complete metric spaces are ...
.


Exterior of a set

The exterior of a subset S of a topological space X, denoted by \operatorname_X S or simply \operatorname S, is the largest open set disjoint from S, namely, it is the union of all open sets in X that are disjoint from S. The exterior is the interior of the complement, which is the same as the complement of the closure; in formulas, \operatornameS = \operatorname(X\setminus S) = X\setminus\overline. Similarly, the interior is the exterior of the complement: \operatornameS = \operatorname(X \setminus S). The interior,
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
, and exterior of a set S together
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
the whole space into three blocks (or fewer when one or more of these is empty): X = \operatornameS \cup \partial S \cup \operatornameS, where \partial S denotes the boundary of S. The interior and exterior are always
open Open or OPEN may refer to: Music * Open (band), Australian pop/rock band * The Open (band), English indie rock band * ''Open'' (Blues Image album), 1969 * ''Open'' (Gotthard album), 1999 * ''Open'' (Cowboy Junkies album), 2001 * ''Open'' (YF ...
, while the boundary is
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
. Some of the properties of the exterior operator are unlike those of the interior operator: * The exterior operator reverses inclusions; if S \subseteq T, then \operatornameT \subseteq \operatornameS. * The exterior operator is not
idempotent Idempotence (, ) is the property of certain operation (mathematics), operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence ...
. It does have the property that \operatornameS \subseteq \operatorname\left(\operatornameS\right).


Interior-disjoint shapes

Two shapes and 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

* * * * * *


References


Bibliography

* * * * * * * * * *


External links

* {{Topology, expanded Closure operators General topology