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 ...
, open sets are a
generalization A generalization is a form of abstraction Abstraction in its main sense is a conceptual process where general rules and concept Concepts are defined as abstract ideas or general notions that occur in the mind, in speech, or in thought. They ...

generalization
of
open interval 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 ge ...
s in the
real line 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 ...
. In a
metric 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 gene ...
—that is, when a
distance function 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 ge ...
is defined—open sets are the sets that, with every point , contain all points that are sufficiently near to (that is, all points whose distance to is less than some value depending on ). More generally, one defines open sets as the members of a given collection of subsets of a given set, a collection that has the property of containing every union of its members, every finite
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
of its members, the
empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

empty set
, and the whole set itself. A set in which such a collection is given is called 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 ...
, and the collection is called a
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 ...
. These conditions are very loose, and allow enormous flexibility in the choice of open sets. For example, ''every'' subset can be open (the
discrete topology In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
), or no set can be open except the space itself and the empty set (the
indiscrete topologyIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
). In practice, however, open sets are usually chosen to provide a notion of nearness that is similar to that of metric spaces, without having a notion of distance defined. In particular, a topology allows defining properties such as continuity,
connectedness In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
, and
compactness In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

compactness
, which were originally defined by means of a distance. The most common case of a topology without any distance is given by
manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ...

manifold
s, which are topological spaces that, ''near'' each point, resemble an open set of a
Euclidean space Euclidean space is the fundamental space of . Originally, it was the of , but in modern there are Euclidean spaces of any nonnegative integer , including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduce ...
, but on which no distance is defined in general. Less intuitive topologies are used in other branches of mathematics; for example, the
Zariski topology In algebraic geometry Algebraic geometry is a branch of mathematics, classically studying zero of a function, zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commut ...
, which is fundamental in
algebraic geometry Algebraic geometry is a branch of 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 thei ...

algebraic geometry
and
scheme theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
.


Motivation

Intuitively, an open set provides a method to distinguish two
points Point or points may refer to: Places * Point, Lewis, a peninsula in the Outer Hebrides, Scotland * Point, Texas, a city in Rains County, Texas, United States * Point, the NE tip and a ferry terminal of Lismore, Scotland, Lismore, Inner Hebrides, ...
. For example, if about one of two points in 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 ...
, there exists an open set not containing the other (distinct) point, the two points are referred to as
topologically distinguishable In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical struct ...
. In this manner, one may speak of whether two points, or more generally two
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
s, of a topological space are "near" without concretely defining a
distance Distance is a numerical measurement Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and ...
. Therefore, topological spaces may be seen as a generalization of spaces equipped with a notion of distance, which are called
metric 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 gene ...
s. In the set of all
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, one has the natural Euclidean metric; that is, a function which measures the distance between two real numbers: . Therefore, given a real number ''x'', one can speak of the set of all points close to that real number; that is, within ''ε'' of ''x''. In essence, points within ε of ''x'' approximate ''x'' to an accuracy of degree ''ε''. Note that ''ε'' > 0 always but as ''ε'' becomes smaller and smaller, one obtains points that approximate ''x'' to a higher and higher degree of accuracy. For example, if ''x'' = 0 and ''ε'' = 1, the points within ''ε'' of ''x'' are precisely the points of the interval (−1, 1); that is, the set of all real numbers between −1 and 1. However, with ''ε'' = 0.5, the points within ''ε'' of ''x'' are precisely the points of (−0.5, 0.5). Clearly, these points approximate ''x'' to a greater degree of accuracy than when ''ε'' = 1. The previous discussion shows, for the case ''x'' = 0, that one may approximate ''x'' to higher and higher degrees of accuracy by defining ''ε'' to be smaller and smaller. In particular, sets of the form (−''ε'', ''ε'') give us a lot of information about points close to ''x'' = 0. Thus, rather than speaking of a concrete Euclidean metric, one may use sets to describe points close to ''x''. This innovative idea has far-reaching consequences; in particular, by defining different collections of sets containing 0 (distinct from the sets (−''ε'', ''ε'')), one may find different results regarding the distance between 0 and other real numbers. For example, if we were to define R as the only such set for "measuring distance", all points are close to 0 since there is only one possible degree of accuracy one may achieve in approximating 0: being a member of R. Thus, we find that in some sense, every real number is distance 0 away from 0. It may help in this case to think of the measure as being a binary condition: all things in R are equally close to 0, while any item that is not in R is not close to 0. In general, one refers to the family of sets containing 0, used to approximate 0, as a ''neighborhood basis''; a member of this neighborhood basis is referred to as an open set. In fact, one may generalize these notions to an arbitrary set (''X''); rather than just the real numbers. In this case, given a point (''x'') of that set, one may define a collection of sets "around" (that is, containing) ''x'', used to approximate ''x''. Of course, this collection would have to satisfy certain properties (known as axioms) for otherwise we may not have a well-defined method to measure distance. For example, every point in ''X'' should approximate ''x'' to ''some'' degree of accuracy. Thus ''X'' should be in this family. Once we begin to define "smaller" sets containing ''x'', we tend to approximate ''x'' to a greater degree of accuracy. Bearing this in mind, one may define the remaining axioms that the family of sets about ''x'' is required to satisfy.


Definitions

Several definitions are given here, in an increasing order of technicality. Each one is a special case of the next one.


Euclidean space

A subset U of the Euclidean -space is ''open'' if, for every point in U,
there exists In predicate logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal system A formal system is an used for inferring theorems from axioms according to a s ...
a positive real number (depending on ) such that a point in belongs to U as soon as its
Euclidean distance 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 an ...
from is smaller than . Equivalently, a subset U of is open if every point in U is the center of 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 ...
contained in U.


Metric space

A subset ''U'' of a
metric 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 gene ...
is called ''open'' if, given any point ''x'' in ''U'', there exists a real number ''ε'' > 0 such that, given any point y \in M satisfying , ''y'' also belongs to ''U''. Equivalently, ''U'' is open if every point in ''U'' has a neighborhood contained in ''U''. This generalizes the Euclidean space example, since Euclidean space with the Euclidean distance is a metric space.


Topological space

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 a set on which a
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 ...
is defined, which consists of a collection of subsets that are said to be ''open'', and satisfy the axioms given below. More precisely, let X be a set. A family \tau of subsets of X is a ''topology'' on X, and the elements of \tau are the ''open sets'' of the topology if * X \in \tau and \varnothing \in \tau\qquad\qquad\qquad (both X and \varnothing are open sets) * \left\ \subseteq \tau then \bigcup_ U_i \in \tau\qquad (any union of open sets is an open set) * U_1, \ldots, U_n \in \tau then U_1 \cap \cdots \cap U_n \in \tau\qquad (any finite intersection of open sets is an open set) Infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form \left( -1/n, 1/n \right), where n is a positive integer, is the set \ which is not open in the real line. A metric space is a topological space, whose topology consists of the collection of all subsets that are unions of open balls. There are, however, topological spaces that are not metric spaces.


Special types of open sets


Clopen sets and non-open and/or non-closed sets

A set might be open, closed, both, or neither. In particular, open and closed sets are not mutually exclusive, meaning that it is in general possible for a subset of a topological space to simultaneously be both an open subset a closed subset. Such subsets are known as . Explicitly, a subset S of a topological space (X, \tau) is called if both S and its complement X \setminus S are open subsets of (X, \tau); or equivalently, if S \in \tau and X \setminus S \in \tau. In topological space (X, \tau), the empty set \varnothing and the set X itself are always open. These two sets are the most well-known examples of clopen subsets and they show that clopen subsets exist in topological space. To see why X is clopen, begin by recalling that the sets X and \varnothing are, by definition, always open subsets (of X). Also by definition, a subset S is called if (and only if) its complement in X, which is the set X \setminus S, is an open subset. Because the complement (in X) of the entire set S := X is the empty set (i.e. X \setminus S = \varnothing), which is an open subset, this means that S = X is a closed subset of X (by definition of "closed subset"). Hence, no matter what topology is placed on X, the entire space X is simultaneously both an open subset and also a closed subset of X; said differently, X is a clopen subset of X. Because the empty set's complement is X \setminus \varnothing = X, which is an open subset, the same reasoning can be used to conclude that S := \varnothing is also a clopen subset of X. Consider the real line \R endowed with its usual
Euclidean topologyIn mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean distance, Euclidean metric. Definition In any metric space, the Ball (mathematics), ope ...
, whose open sets are defined as follows: every interval (a, b) of real numbers belongs to the topology, every union of such intervals, e.g. (a, b) \cup (c, d), belongs to the topology, and as always, both \R and \varnothing belong to the topology. * The interval I = (0, 1) is open in \R because it belongs to the Euclidean topology. If I were to have an open complement, it would mean by definition that I were closed. But I does not have an open complement; its complement is \R \setminus I = (-\infty, 0] \cup , \infty), which does belong to the Euclidean topology since it is not a union of Interval (mathematics)#Including or excluding endpoints, open intervals of the form (a, b). Hence, I is an example of a set that is open but not closed. * By a similar argument, the interval J = [0, 1] is a closed subset but not an open subset. * Finally, since neither K = [0, 1) nor its complement \R \setminus K = (-\infty, 0) \cup [1, \infty) belongs to the Euclidean topology (because it can not be written as a union of intervals of the form (a, b)), this means that K is neither open nor closed. If a topological space X is endowed with the
discrete topology In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
(so that by definition, every subset of X is open) then every subset of X is a clopen subset. For a more advanced example reminiscent of the discrete topology, suppose that \mathcal is an
ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (poset) ''P'' is a certain subset of ''P,'' namely a maximal filter on ''P'', that is, a proper filter on ''P'' that cannot be enlarged to a bigger pr ...
on a non-empty set X. Then the union \tau := \mathcal \cup \ is a topology on X with the property that non-empty proper subset S of X is an open subset or else a closed subset, but never both; that is, if \varnothing \neq S \subsetneq X (where S \neq X) then of the following two statements is true: either (1) S \in \tau or else, (2) X \setminus S \in \tau. Said differently, subset is open or closed but the subsets that are both (i.e. that are clopen) are \varnothing and X.


Regular open sets

A subset S of a topological space X is called a if \operatorname \left( \overline \right) = S or equivalently, if \operatorname \left( \overline \right) = \operatorname S, where \operatorname S (resp. \operatorname S, \overline) denotes the topological boundary (resp.
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
, closure) of S in X. A topological space for which there exists a
base Base or BASE may refer to: Brands and enterprises * Base (mobile telephony provider), a Belgian mobile telecommunications operator *Base CRM Base CRM (originally Future Simple or PipeJump) is an enterprise software company based in Mountain Vie ...
consisting of regular open sets is called a . A subset of X is a regular open set if and only if its complement in X is a regular closed set, where by definition a subset S of X is called a if \overline = S or equivalently, if \operatorname \left( \operatorname S \right) = \operatorname S. Every regular open set (resp. regular closed set) is an open subset (resp. is a closed subset) although in general,One exception if the if X is endowed with the
discrete topology In topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric objec ...
, in which case every subset of X is both a regular open subset and a regular closed subset of X.
the converses are true.


Properties

The union of any number of open sets, or infinitely many open sets, is open. The
intersection The line (purple) in two points (red). The disk (yellow) intersects the line in the line segment between the two red points. In mathematics, the intersection of two or more objects is another, usually "smaller" object. Intuitively, the inter ...
of a finite number of open sets is open. A
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 ...
of an open set (relative to the space that the topology is defined on) is called a
closed set In geometry Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...
. A set may be both open and closed (a
clopen set In topology, a clopen set (a portmanteau of closed-open set) in a topological space is a set which is both open set, open and closed set, closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but ...
). The
empty set #REDIRECT Empty set#REDIRECT Empty set In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, cha ...

empty set
and the full space are examples of sets that are both open and closed.


Uses

Open sets have a fundamental importance 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 ...

topology
. The concept is required to define and make sense of
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 ...
and other topological structures that deal with the notions of closeness and convergence for spaces such as
metric spaces Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement Mathematics * Metric (mathematics), an abstraction of the notion of ''distance'' in a metric space * Metric tensor, in differential geometr ...
and
uniform spaces In the mathematical field of topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathemati ...
. Every
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
''A'' of a topological space ''X'' contains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called the
interior Interior may refer to: Arts and media * Interior (Degas), ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * Interior (play), ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * The Interior (novel) ...
of ''A''. It can be constructed by taking the union of all the open sets contained in ''A''. A
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
f : X \to Y between two topological spaces X and Y is if the
preimage In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
of every open set in Y is open in X. The function f : X \to Y is called if the
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of every open set in X is open in Y. An open set on the
real line 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 ...
has the characteristic property that it is a countable union of disjoint open intervals.


Notes and cautions


"Open" is defined relative to a particular topology

Whether a set is open depends on the
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 ...

topology
under consideration. Having opted for greater brevity over greater clarity, we refer to a set ''X'' endowed with a topology \tau as "the topological space ''X''" rather than "the topological space (X, \tau)", despite the fact that all the topological data is contained in \tau. If there are two topologies on the same set, a set ''U'' that is open in the first topology might fail to be open in the second topology. For example, if ''X'' is any topological space and ''Y'' is any subset of ''X'', the set ''Y'' can be given its own topology (called the 'subspace topology') defined by "a set ''U'' is open in the subspace topology on ''Y'' if and only if ''U'' is the intersection of ''Y'' with an open set from the original topology on ''X''." This potentially introduces new open sets: if ''V'' is open in the original topology on ''X'', but V \cap Y isn't open in the original topology on ''X'', then V \cap Y is open in the subspace topology on ''Y''. As a concrete example of this, if ''U'' is defined as the set of rational numbers in the interval (0, 1), then ''U'' is an open subset of the
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, but not of the
real numbers 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 an ...

real numbers
. This is because when the surrounding space is the rational numbers, for every point ''x'' in ''U'', there exists a positive number ''a'' such that all points within distance ''a'' of ''x'' are also in ''U''. On the other hand, when the surrounding space is the reals, then for every point ''x'' in ''U'' there is positive ''a'' such that all points within distance ''a'' of ''x'' are in ''U'' (because ''U'' contains no non-rational numbers).


Generalizations of open sets

Throughout, (X, \tau) will be a topological space. A subset A \subseteq X of a topological space X is called: Using the fact that :A ~\subseteq~ \operatorname_X A ~\subseteq~ \operatorname_X B and \operatorname_X A ~\subseteq~ \operatorname_X B ~\subseteq~ B whenever two subsets A, B \subseteq X satisfy A \subseteq B, the following may be deduced: * Every α-open subset is semi-open, semi-preopen, preopen, and b-open. * Every b-open set is semi-preopen (i.e. β-open). * Every preopen set is b-open and semi-preopen. * Every semi-open set is b-open and semi-preopen. Moreover, a subset is a regular open set if and only if it is preopen and semi-closed. The intersection of an α-open set and a semi-preopen (resp. semi-open, preopen, b-open) set is a semi-preopen (resp. semi-open, preopen, b-open) set. Preopen sets need not be semi-open and semi-open sets need not be preopen. Arbitrary unions of preopen (resp. α-open, b-open, semi-preopen) sets are once again preopen (resp. α-open, b-open, semi-preopen). However, finite intersections of preopen sets need not be preopen. The set of all α-open subsets of a space (X, \tau) forms a topology on X that is finer than \tau. A topological space X is if and only if every compact subspace of X is θ-closed. A space X is
totally disconnectedIn topology s, which have only one surface and one edge, are a kind of object studied in topology. In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...
if and only if every regular closed subset is preopen or equivalently, if every semi-open subset is preopen. Moreover, the space is totally disconnected if and only if the of every preopen subset is open.


See also

* * * * * * *


Notes


References


Bibliography

* *


External links

* {{Topology General topology