Real line
Encyclopedia
In mathematics
, the real line, or real number line is the line
whose points
are the real number
s. That is, the real line is the set of all real numbers, viewed as a geometric
space, namely the Euclidean space
of dimension
one. It can be thought of as a vector space
(or affine space
), a metric space
, a topological space
, a measure space, or a linear continuum
.
Just like the set of real numbers, the real line is usually denoted by the symbol (or alternatively,
, the letter “R
” in blackboard bold
). However, it is sometimes denoted in order to emphasize its role as the first Euclidean space.
This article focuses on the aspects of as a geometric space in topology
, geometry, and real analysis
. The real numbers also play an important role in algebra
as a field
, but in this context is rarely referred to as a line. For more information on in all of its guises, see real number
.
under the standard ordering. Specifically, the real line is linearly ordered by , and this ordering is dense
and has the least-upper-bound property
.
In addition to the above properties, the real line has no maximum
or minimum element. It also has a countable
dense
subset
, namely the set of rational number
s. It is a theorem that any linear continuum with a countable dense subset and no maximum or minimum element is order-isomorphic
to the real line.
The real line also satisfies the countable chain condition
: every collection of mutually disjoint, nonempty open interval
s in is countable. In order theory
, the famous Suslin problem
asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order-isomorphic to . This statement has been shown to be independent
of the standard axiomatic system of set theory
known as ZFC.
If and , then the -ball
in centered at is simply the open interval
.
This real line has several important properties as a metric space:
The real line carries a standard topology
which can be introduced in two different, equivalent ways.
First, since the real numbers are totally ordered
, they carry an order topology
. Second, the real numbers inherit a metric topology from the metric defined above. The order topology on metric topology on are the same. As a topological space, the real line is homeomorphic
to the open interval .
The real line is trivially a topological manifold
of dimension
. Up to homeomorphism, it is one of only two different 1-manifolds without boundary, the other being the circle
. It also has a standard differentiable structure on it, making it a differentiable manifold
. (Up to diffeomorphism
, there is only one differentiable structure that the topological space supports.)
The real line is locally compact and paracompact, as well as second-countable
and normal
. It is also path-connected, and is therefore connected
as well, though it can be disconnected by removing any one point. The real line is also contractible, and as such all of its homotopy group
s and reduced homology
groups are zero.
As a locally compact space
, the real line can be compactified in several different ways. The one-point compactification of is a circle (namely the real projective line), and the extra point can be thought of as an unsigned infinity. Alternatively, the real line has two ends, and the resulting end compactification is the extended real line
. There is also the Stone–Čech compactification
of the real line, which involves adding an infinite number of additional points.
In some contexts, it is helpful to place other topologies on the set of real numbers, such as the lower limit topology
or the Zariski topology
. For the real numbers, the latter is the same as the finite complement topology.
over the field
of real numbers (that is, over itself) of dimension
. It has a standard inner product, making it a Euclidean space
. (The inner product is simply ordinary multiplication
of real numbers.) The standard norm
on is simply the absolute value
function.
, namely the Lebesgue measure
. This measure can be defined as the completion
of a Borel measure defined on , where the measure of any interval is the length of the interval.
Lebesgue measure on the real line is one of the simplest examples of a Haar measure
on a locally compact group
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the real line, or real number line is the line
Line (geometry)
The notion of line or straight line was introduced by the ancient mathematicians to represent straight objects with negligible width and depth. Lines are an idealization of such objects...
whose points
Point (geometry)
In geometry, topology and related branches of mathematics a spatial point is a primitive notion upon which other concepts may be defined. In geometry, points are zero-dimensional; i.e., they do not have volume, area, length, or any other higher-dimensional analogue. In branches of mathematics...
are the real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s. That is, the real line is the set of all real numbers, viewed as a geometric
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
space, namely the Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
of dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
one. It can be thought of as a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
(or affine space
Affine space
In mathematics, an affine space is a geometric structure that generalizes the affine properties of Euclidean space. In an affine space, one can subtract points to get vectors, or add a vector to a point to get another point, but one cannot add points. In particular, there is no distinguished point...
), a metric space
Metric space
In mathematics, a metric space is a set where a notion of distance between elements of the set is defined.The metric space which most closely corresponds to our intuitive understanding of space is the 3-dimensional Euclidean space...
, a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, a measure space, or a linear continuum
Linear continuum
In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line.Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two members there is another, and which "lacks gaps" in the...
.
Just like the set of real numbers, the real line is usually denoted by the symbol (or alternatively,
, the letter “RR
R is the eighteenth letter of the basic modern Latin alphabet.-History:The original Semitic letter may have been inspired by an Egyptian hieroglyph for tp, "head". It was used for by Semites because in their language, the word for "head" was rêš . It developed into Greek Ρ and Latin R...
” in blackboard bold
Blackboard bold
Blackboard bold is a typeface style that is often used for certain symbols in mathematical texts, in which certain lines of the symbol are doubled. The symbols usually denote number sets...
). However, it is sometimes denoted in order to emphasize its role as the first Euclidean space.
This article focuses on the aspects of as a geometric space in topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, geometry, and real analysis
Real analysis
Real analysis, is a branch of mathematical analysis dealing with the set of real numbers and functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real...
. The real numbers also play an important role in algebra
Algebra
Algebra is the branch of mathematics concerning the study of the rules of operations and relations, and the constructions and concepts arising from them, including terms, polynomials, equations and algebraic structures...
as a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, but in this context is rarely referred to as a line. For more information on in all of its guises, see real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
.
As a linear continuum
The real line is a linear continuumLinear continuum
In the mathematical field of order theory, a continuum or linear continuum is a generalization of the real line.Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two members there is another, and which "lacks gaps" in the...
under the standard ordering. Specifically, the real line is linearly ordered by , and this ordering is dense
Dense order
In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x In mathematics, a partial order ≤ on a set X is said to be dense if, for all x and y in X for which x...
and has the least-upper-bound property
Least-upper-bound property
In mathematics, the least-upper-bound property is a fundamental property of the real numbers and certain other ordered sets. The property states that any non-empty set of real numbers that has an upper bound necessarily has a least upper bound ....
.
In addition to the above properties, the real line has no maximum
Greatest element
In mathematics, especially in order theory, the greatest element of a subset S of a partially ordered set is an element of S which is greater than or equal to any other element of S. The term least element is defined dually...
or minimum element. It also has a countable
Countable set
In mathematics, a countable set is a set with the same cardinality as some subset of the set of natural numbers. A set that is not countable is called uncountable. The term was originated by Georg Cantor...
dense
Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense if any point x in X belongs to A or is a limit point of A...
subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
, namely the set of rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s. It is a theorem that any linear continuum with a countable dense subset and no maximum or minimum element is order-isomorphic
Order isomorphism
In the mathematical field of order theory an order isomorphism is a special kind of monotone function that constitutes a suitable notion of isomorphism for partially ordered sets . Whenever two posets are order isomorphic, they can be considered to be "essentially the same" in the sense that one of...
to the real line.
The real line also satisfies the countable chain condition
Countable chain condition
In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable. There are really two conditions: the upwards and downwards countable chain conditions. These are not equivalent...
: every collection of mutually disjoint, nonempty open interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
s in is countable. In order theory
Order theory
Order theory is a branch of mathematics which investigates our intuitive notion of order using binary relations. It provides a formal framework for describing statements such as "this is less than that" or "this precedes that". This article introduces the field and gives some basic definitions...
, the famous Suslin problem
Suslin's problem
In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin in a work published posthumously in 1920....
asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order-isomorphic to . This statement has been shown to be independent
Independence (mathematical logic)
In mathematical logic, independence refers to the unprovability of a sentence from other sentences.A sentence σ is independent of a given first-order theory T if T neither proves nor refutes σ; that is, it is impossible to prove σ from T, and it is also impossible to prove from T that...
of the standard axiomatic system of set theory
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
known as ZFC.
As a metric space
The real line forms a metric space, with the metric given by absolute difference:- .
If and , then the -ball
Ball (mathematics)
In mathematics, a ball is the space inside a sphere. It may be a closed ball or an open ball ....
in centered at is simply the open interval
Interval (mathematics)
In mathematics, a interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers satisfying is an interval which contains and , as well as all numbers between them...
.
This real line has several important properties as a metric space:
- The real line is a complete metric space, in the sense that any Cauchy sequenceCauchy sequenceIn mathematics, a Cauchy sequence , named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses...
of points converges. - The real line is path-connected, and is one of the simplest examples of a geodesic metric space
- The Hausdorff dimensionHausdorff dimensionthumb|450px|Estimating the Hausdorff dimension of the coast of Great BritainIn mathematics, the Hausdorff dimension is an extended non-negative real number associated with any metric space. The Hausdorff dimension generalizes the notion of the dimension of a real vector space...
of the real line is equal to one. - The isometry groupIsometry groupIn mathematics, the isometry group of a metric space is the set of all isometries from the metric space onto itself, with the function composition as group operation...
of the real line, also known as the Euclidean groupEuclidean groupIn mathematics, the Euclidean group E, sometimes called ISO or similar, is the symmetry group of n-dimensional Euclidean space...
, consists of all functions of the form , where is a real number. This group is isomorphicGroup isomorphismIn abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic...
to a semidirect productSemidirect productIn mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
of the additive group of with a cyclic groupCyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
of order two, and is an example of a generalized dihedral groupGeneralized dihedral groupIn mathematics, the generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, and the orthogonal group O.-Definition:...
.
As a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
which can be introduced in two different, equivalent ways.
First, since the real numbers are totally ordered
Total order
In set theory, a total order, linear order, simple order, or ordering is a binary relation on some set X. The relation is transitive, antisymmetric, and total...
, they carry an order topology
Order topology
In mathematics, an order topology is a certain topology that can be defined on any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets...
. Second, the real numbers inherit a metric topology from the metric defined above. The order topology on metric topology on are the same. As a topological space, the real line is homeomorphic
Homeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
to the open interval .
The real line is trivially a topological manifold
Topological manifold
In mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
of dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
. Up to homeomorphism, it is one of only two different 1-manifolds without boundary, the other being the circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
. It also has a standard differentiable structure on it, making it a differentiable manifold
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
. (Up to diffeomorphism
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds. It is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth.- Definition :...
, there is only one differentiable structure that the topological space supports.)
The real line is locally compact and paracompact, as well as second-countable
Second-countable space
In topology, a second-countable space, also called a completely separable space, is a topological space satisfying the second axiom of countability. A space is said to be second-countable if its topology has a countable base...
and normal
Normal space
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space...
. It is also path-connected, and is therefore connected
Connected space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
as well, though it can be disconnected by removing any one point. The real line is also contractible, and as such all of its homotopy group
Homotopy group
In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space...
s and reduced homology
Reduced homology
In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, designed to make a point have all its homology groups zero...
groups are zero.
As a locally compact space
Locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.-Formal definition:...
, the real line can be compactified in several different ways. The one-point compactification of is a circle (namely the real projective line), and the extra point can be thought of as an unsigned infinity. Alternatively, the real line has two ends, and the resulting end compactification is the extended real line
Extended real number line
In mathematics, the affinely extended real number system is obtained from the real number system R by adding two elements: +∞ and −∞ . The projective extended real number system adds a single object, ∞ and makes no distinction between "positive" or "negative" infinity...
. There is also the Stone–Čech compactification
Stone–Cech compactification
In the mathematical discipline of general topology, Stone–Čech compactification is a technique for constructing a universal map from a topological space X to a compact Hausdorff space βX...
of the real line, which involves adding an infinite number of additional points.
In some contexts, it is helpful to place other topologies on the set of real numbers, such as the lower limit topology
Lower limit topology
In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R of real numbers; it is different from the standard topology on R and has a number of interesting properties...
or the Zariski topology
Zariski topology
In algebraic geometry, the Zariski topology is a particular topology chosen for algebraic varieties that reflects the algebraic nature of their definition. It is due to Oscar Zariski and took a place of particular importance in the field around 1950...
. For the real numbers, the latter is the same as the finite complement topology.
As a vector space
The real line is a vector spaceVector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over the field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
of real numbers (that is, over itself) of dimension
Dimension
In physics and mathematics, the dimension of a space or object is informally defined as the minimum number of coordinates needed to specify any point within it. Thus a line has a dimension of one because only one coordinate is needed to specify a point on it...
. It has a standard inner product, making it a Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
. (The inner product is simply ordinary multiplication
Multiplication
Multiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
of real numbers.) The standard norm
Norm (mathematics)
In linear algebra, functional analysis and related areas of mathematics, a norm is a function that assigns a strictly positive length or size to all vectors in a vector space, other than the zero vector...
on is simply the absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
function.
As a measure space
The real line carries a canonical measureMeasure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign to each suitable subset a number, intuitively interpreted as the size of the subset. In this sense, a measure is a generalization of the concepts of length, area, and volume...
, namely the Lebesgue measure
Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space. For n = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called...
. This measure can be defined as the completion
Complete measure
In mathematics, a complete measure is a measure space in which every subset of every null set is measurable...
of a Borel measure defined on , where the measure of any interval is the length of the interval.
Lebesgue measure on the real line is one of the simplest examples of a Haar measure
Haar measure
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
on a locally compact group
Locally compact group
In mathematics, a locally compact group is a topological group G which is locally compact as a topological space. Locally compact groups are important because they have a natural measure called the Haar measure. This allows one to define integrals of functions on G.Many of the results of finite...
.