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 ...
, open sets are a
generalization
A generalization is a form of abstraction whereby common properties of specific instances are formulated as general concepts or claims. Generalizations posit the existence of a domain or set of elements, as well as one or more common characte ...
of
open interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
s in 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 ...
.
In 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 ...
(a
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
along with a
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
defined between any two points), 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
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 its members, every finite
intersection of its members, the
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
, and the whole set itself. A set in which such a collection is given is called 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 ...
, and the collection is called a
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 ...
. 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, 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 ...
), or no set can be open except the space itself and the empty set (the
indiscrete topology 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 ...
).
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, connectedness is used to refer to various properties meaning, in some sense, "all one piece". When a mathematical object has such a property, we say it is connected; otherwise it is disconnected. When a disconnected object can be s ...
, and
compactness, which were originally defined by means of a distance.
The most common case of a topology without any distance is given by
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
s, which are topological spaces that, ''near'' each point, resemble an open set 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 ...
, 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 and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
, which is fundamental in
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical ...
and
scheme theory
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different sc ...
.
Motivation
Intuitively, an open set provides a method to distinguish two
points. For example, if about one of two points in 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 ...
, there exists an open set not containing the other (distinct) point, the two points are referred to as
topologically distinguishable
In topology, two points of a topological space ''X'' are topologically indistinguishable if they have exactly the same neighborhoods. That is, if ''x'' and ''y'' are points in ''X'', and ''Nx'' is the set of all neighborhoods that contain ''x'', ...
. In this manner, one may speak of whether two points, or more generally two
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 ...
s, of a topological space are "near" without concretely defining a
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...
. Therefore, topological spaces may be seen as a generalization of spaces equipped with a notion of distance, which are called
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 ...
s.
In the set of all
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 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
of the
Euclidean -space is ''open'' if, for every point in
,
there exists
In predicate logic, an existential quantification is a type of quantifier, a logical constant which is interpreted as "there exists", "there is at least one", or "for some". It is usually denoted by the logical operator symbol ∃, which, w ...
a positive real number (depending on ) such that any point in whose
Euclidean distance
In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points.
It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
from is smaller than belongs to
. Equivalently, a subset
of is open if every point in
is the center of 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 defi ...
contained in
An example of a subset of that is not open is the
closed interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
, since neither nor belongs to for any , no matter how small.
Metric space
A subset ''U'' 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 ...
is called ''open'' if, for any point ''x'' in ''U'', there exists a real number ''ε'' > 0 such that any point
satisfying 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
''topology'' on a set is a set of subsets of with the properties below. Each member of
is called an ''open set''.
*
and
*Any union of sets in
belong to
: if
then
*Any finite intersection of sets in
belong to
: if
then
together with
is called 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 ...
.
Infinite intersections of open sets need not be open. For example, the intersection of all intervals of the form
where
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
of a topological space
is called if both
and its complement
are open subsets of
; or equivalently, if
and
In topological space
the empty set
and the set
itself are always clopen. 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
is clopen, begin by recalling that the sets
and
are, by definition, always open subsets (of
). Also by definition, a subset
is called if (and only if) its complement in
which is the set
is an open subset. Because the complement (in
) of the entire set
is the empty set (i.e.
), which is an open subset, this means that
is a closed subset of
(by definition of "closed subset"). Hence, no matter what topology is placed on
the entire space
is simultaneously both an open subset and also a closed subset of
; said differently,
is a clopen subset of
Because the empty set's complement is
which is an open subset, the same reasoning can be used to conclude that
is also a clopen subset of
Consider the real line
endowed with its usual
Euclidean topology
In 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
The Euclidean norm on \R^n is the non-negative f ...
, whose open sets are defined as follows: every interval
of real numbers belongs to the topology, every union of such intervals, e.g.
belongs to the topology, and as always, both
and
belong to the topology.
* The interval
is open in
because it belongs to the Euclidean topology. If
were to have an open complement, it would mean by definition that
were closed. But
does not have an open complement; its complement is
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
Hence,
is an example of a set that is open but not closed.
* By a similar argument, the interval
is a closed subset but not an open subset.
* Finally, since neither
nor its complement
belongs to the Euclidean topology (because it can not be written as a union of intervals of the form
), this means that
is neither open nor closed.
If a topological space
is endowed with the
discrete topology
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 ...
(so that by definition, every subset of
is open) then every subset of
is a clopen subset.
For a more advanced example reminiscent of the discrete topology, suppose that
is an
ultrafilter
In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "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 proper filter o ...
on a non-empty set
Then the union
is a topology on
with the property that non-empty proper subset
of
is an open subset or else a closed subset, but never both; that is, if
(where
) then of the following two statements is true: either (1)
or else, (2)
Said differently, subset is open or closed but the subsets that are both (i.e. that are clopen) are
and
Regular open sets
A subset
of a topological space
is called a if
or equivalently, if
where
(resp.
) denotes the
topological boundary
In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term bound ...
(resp.
interior,
closure) of
in
A topological space for which there exists a
base consisting of regular open sets is called a .
A subset of
is a regular open set if and only if its complement in
is a regular closed set, where by definition a subset
of
is called a if
or equivalently, if
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 is endowed with the ]discrete topology
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 ...
, in which case every subset of is both a regular open subset and a regular closed subset of the converses are true.
Properties
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 any number of open sets, or infinitely many open sets, is open.
The
intersection of a finite number of open sets is open.
A
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-clas ...
of an open set (relative to the space that the topology is defined on) is called a
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a cl ...
. 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 and closed. That this is possible may seem counter-intuitive, as the common meanings of and are antonyms, but their mathematical de ...
). The
empty set
In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other ...
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, 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 concept is required to define and make sense of
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 ...
and other topological structures that deal with the notions of closeness and convergence for spaces such as
metric spaces
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 ...
and
uniform spaces
In the mathematical field of topology, a uniform space is a set with a uniform structure. Uniform spaces are topological spaces with additional structure that is used to define uniform properties such as completeness, uniform continuity and unifor ...
.
Every
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 ...
''A'' of a topological space ''X'' contains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called the
interior 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 type of key on computer keyboards
* Function model, a structured representation of processes in a system
* Function object or functor or functionoid, a concept of object-oriente ...
between two topological spaces
and
is if the
preimage of every open set in
is open in
The function
is called if the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensiona ...
of every open set in
is open in
An open set on 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 ...
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, 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 ...
under consideration. Having opted for
greater brevity over greater clarity, we refer to a set ''X'' endowed with a topology
as "the topological space ''X''" rather than "the topological space
", despite the fact that all the topological data is contained in
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
isn't open in the original topology on ''X'', then
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
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 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, but not of the
real numbers
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 ...
. 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,
will be a topological space.
A subset
of a topological space
is called:
- if , and the complement of such a set is called .
- , , or if it satisfies any of the following equivalent conditions:
- There exists subsets such that is open in is a
dense subset
In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the ra ...
of and
- There exists an open (in ) subset such that is a dense subset of
The complement of a preopen set is called .
- if . The complement of a b-open set is called .
- or if it satisfies any of the following equivalent conditions:
- is a regular closed subset of
- There exists a preopen subset of such that
The complement of a β-open set is called .
- if it satisfies any of the following equivalent conditions:
- Whenever a sequence in converges to some point of then that sequence is eventually in Explicitly, this means that if is a sequence in and if there exists some is such that in then is eventually in (that is, there exists some integer such that if then ).
- is equal to its in which by definition is the set
:
The complement of a sequentially open set is called . A subset is sequentially closed in if and only if is equal to its , which by definition is the set consisting of all for which there exists a sequence in that converges to (in ).
- and is said to have if there exists an open subset such that is a meager subset, where denotes the
symmetric difference
In mathematics, the symmetric difference of two sets, also known as the disjunctive union, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets \ and \ is \.
Th ...
.[.]
* The subset is said to have the Baire property in the restricted sense if for every subset of the intersection has the Baire property relative to .[.]
- if . The complement in of a semi-open set is called a set.
* The (in ) of a subset denoted by is the intersection of all semi-closed subsets of that contain as a subset.
- if for each there exists some semiopen subset of such that
- (resp. ) if its complement in is a θ-closed (resp. ) set, where by definition, a subset of is called (resp. ) if it is equal to the set of all of its θ-cluster points (resp. δ-cluster points). A point is called a (resp. a ) of a subset if for every open neighborhood of in the intersection is not empty (resp. is not empty).
Using the fact that
:
and
whenever two subsets
satisfy
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
forms a topology on
that is
finer than
A topological space
is
Hausdorff if and only if every
compact subspace of
is θ-closed.
A space
is
totally disconnected
In topology and related branches of mathematics, a totally disconnected space is a topological space that has only singletons as connected subsets. In every topological space, the singletons (and, when it is considered connected, the empty set) ...
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