HOME

TheInfoList




In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, specifically in
topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
, the interior of a
subset In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

subset
of a
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
is the union of all subsets of that are
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australia ...
in . A point that is in the interior of is an interior point of . The interior of is the
complement A complement is often something that completes something else, or at least adds to it in some useful way. Thus it may be: * Complement (linguistics), a word or phrase having a particular syntactic role ** Subject complement, a word or phrase addi ...

complement
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 ...
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
the whole space into three blocks (or fewer when one or more of these is empty). The interior and exterior are always
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australia ...
while the boundary is always closed. Sets with empty interior have been called boundary sets.


Definitions


Interior point

If is a subset of a
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
, then is an interior point of if there exists an
open ball In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
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 Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
with metric : is an interior point of if there exists , such that is in whenever the distance . This definition generalises to
topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
s by replacing "open ball" with "
open set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
". 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, Hiberno-English, Hibernian English, Australian English and Canadian English) or neighborhood (American English; American and British English spelling differences, see spelling differences) is a geographicall ...
of .)


Interior of a set

The interior of a subset of a topological space , denoted by or , 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 , then . *If is the Euclidean space \mathbb of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s, then . *If is the Euclidean space \mathbb, then the interior of the set \mathbb of
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction (mathematics), fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ) ...
s is empty. *If is the
complex plane In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

complex plane
\mathbb = \mathbb^2, then \operatorname(\) = \. *In any Euclidean space, the interior of any
finite set In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...
is the empty set. On the set of real numbers, one can put other topologies rather than the standard one. *If , where \mathbb has the
lower limit topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
, then int(
, 1 The comma is a punctuation Punctuation (or sometimes interpunction) is the use of spacing, conventional signs (called punctuation marks), and certain typographical devices as aids to the understanding and correct reading of written text, ...
= [0, 1). *If one considers on \mathbb the topology in which every set is open, then . *If one considers on \mathbb the topology in which the only open sets are the empty set and \mathbb 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, 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, we have and for every subset, proper subset of , is the empty set.


Properties

Let be a topological space and let and be subset of . * is
open Open or OPEN may refer to: Music * Open (band) Open is a band. Background Drummer Pete Neville has been involved in the Sydney/Australian music scene for a number of years. He has recently completed a Masters in screen music at the Australia ...
in . * If is open in then if and only if . * is an open subset of when is given the
subspace topology In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...

subspace topology
. * is an open subset of
if and only if In logic Logic is an interdisciplinary field which studies truth and reasoning. Informal logic seeks to characterize Validity (logic), valid arguments informally, for instance by listing varieties of fallacies. Formal logic represents st ...
. * : . * : . * /: . * /: If then . The above statements will remain true if all instances of the symbols/words :"interior", "Int", "open", "subset", and "largest" are respectively replaced by :"closure", "Cl", "closed", "superset", and "smallest" and the following symbols are swapped: # "⊆" swapped with "⊇" # "∪" swapped with "∩" For more details on this matter, see
interior operatorIn mathematics, a closure operator on a Set (mathematics), set ''S'' is a Function (mathematics), function \operatorname: \mathcal(S)\rightarrow \mathcal(S) from the power set of ''S'' to itself that satisfies the following conditions for all sets X, ...
below or the article
Kuratowski closure axioms In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...
. Other properties include: * If is closed in and then .


Interior operator

The interior operator \operatorname_X is dual to the 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 Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...
containing S, and the backslash \,\setminus\, denotes
set-theoretic difference In set theory, the complement of a Set (mathematics), set , often denoted by (or ), are the Element (mathematics), elements not in . When all sets under consideration are considered to be subsets of a given set , the absolute complement of is t ...
. Therefore, the abstract theory of closure operators and the
Kuratowski closure axioms In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and ...
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 Analysis is the branch of mathematics dealing with Limit (mathematics), limits and related theories, such as Derivative, differentiation, Integral, integration, Measure (mathematics), measure, sequences, Series (mathema ...
the following result does hold: The result above implies that every complete metric space is a
Baire space In mathematics, a Baire space is a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and ...
.


Exterior of a set

The (topological) exterior of a subset S of a topological space X, denoted by \operatorname_X S or simply \operatorname S, is the complement of the closure of S: :\operatorname_X S := X \setminus \operatorname_X S although it can be equivalently defined in terms of the interior by: :\operatorname_X S = \operatorname_X (X \setminus S) Alternatively, the interior \operatorname_X S could instead be defined in terms of the exterior by using the set equality :\operatorname_X S = \operatorname_X (X \setminus S). As a consequence of this relationship between the interior and exterior, many properties of the exterior \operatorname_X S can be readily deduced directly from those of the interior \operatorname_X S and elementary set identities. Such properties include the following: * \operatorname_X S is an open subset of X that is disjoint from S. * If S \subseteq T then \operatorname_X T \subseteq \operatorname_X S. * \operatorname_X S is equal to the union of all open subsets of X that are disjoint from S. * \operatorname_X S is equal to the largest open subset of X that is disjoint from S. Unlike the interior operator, \operatorname_X is not
idempotent Idempotence (, ) is the property of certain operations Operation or Operations may refer to: Science and technology * Surgical operation Surgery ''cheirourgikē'' (composed of χείρ, "hand", and ἔργον, "work"), via la, chirurgiae, ...
, although it does have the property that \operatorname_X S \subseteq \operatorname_X \left(\operatorname_X S\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 General topology Closure operators