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In
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 words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, the interior of a
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 o ...
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 point ...
is the union 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'' (Y ...
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 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. The interior, boundary, and exterior of a subset together partition the whole space into three blocks (or fewer when one or more of these is empty).

# 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'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
, then is an interior point of if there exists an open ball 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 setti ...
with metric : is an interior point of if there exists $r > 0,$ such that is in whenever the distance $d\left(x, y\right) < r.$ 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 point ...
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 (mathematics), set along with a metric (mathematics), distance defined between any two points), open sets are the sets that, with every ...
". 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 area, ...
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 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 number to a poin ...
$\Reals$ (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 rati ...
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 $\operatorname\left(\\right) = \.$ *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. T ...
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 ...
s, one can put other topologies rather than the standard one: *If is the real numbers $\Reals$ with the lower limit topology, 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 , 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'' (Y ...
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 bico ...
$\operatorname S = S.$ * : $\operatorname S \subseteq S.$ * : $\operatorname \left(\operatorname S\right) = \operatorname S.$ * /: $\operatorname \left(S \cap T\right) = \left(\operatorname S\right) \cap \left(\operatorname T\right).$ ** However, the interior operator does not distribute over unions since only $\operatorname \left(S \cup T\right) ~\supseteq~ \left(\operatorname S\right) \cup \left(\operatorname T\right)$ is guaranteed in general and equality might not hold. For example, if $X = \Reals, S = \left(-\infty, 0\right],$ and $T = \left(0, \infty\right)$ then $\left(\operatorname S\right) \cup \left(\operatorname T\right) = \left(-\infty, 0\right) \cup \left(0, \infty\right) = \Reals \setminus \$ is a proper subset of $\operatorname \left(S \cup T\right) = \operatorname \Reals = \Reals.$ * /: If $S \subseteq T$ then $\operatorname S \subseteq \operatorname T.$ Other properties include: * If is closed in and $\operatorname T = \varnothing$ then $\operatorname \left(S \cup T\right) = \operatorname S.$ 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 below or the article Kuratowski closure axioms.

# 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 point ...
containing $S,$ and the backslash $\,\setminus\,$ denotes 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 $X.$ 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 bou ...
the following result does hold: The result above implies that every complete metric space is a Baire space.

# 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, and exterior of a set $S$ together partition 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'' (Y ...
, while the boundary is closed. 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 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 arises in a number of pl ...
. It does have the property that $\operatornameS \subseteq \operatorname\left\left(\operatornameS\right\right).$

# 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.