Real line

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The real line Real number line.svg
The real line

In mathematics, the real line, or real number line is the line whose points are the real numbers. That is, the real line is the set R 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.

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Just like the set of real numbers, the real line is usually denoted by the symbol R (or alternatively, , the letter “R” in blackboard bold). However, it is sometimes denoted R1 in order to emphasize its role as the first Euclidean space.

This article focuses on the aspects of R 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 R is rarely referred to as a line. For more information on R in all of its guises, see real number.

As a linear continuum

The order on the number line Number line with x smaller than y.svg
The order on the number line
Each set on the real number line has a supremum. Illustration of supremum.svg
Each set on the real number line has a supremum.

The real line is a linear continuum 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 numbers. 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 intervals in R 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 R. This statement has been shown to be independent of the standard axiomatic system of set theory known as ZFC.

As a metric space

The metric on the real line is absolute difference. Absolute difference.svg
The metric on the real line is absolute difference.
An e-ball around a number a Epsilon Umgebung.svg
An ε-ball around a number a

The real line forms a metric space, with the distance function given by absolute difference:

The metric tensor is clearly the 1-dimensional Euclidean metric. Since the n-dimensional Euclidean metric can be represented in matrix form as the n-by-n identity matrix, the metric on the real line is simply the 1-by-1 identity matrix, i.e. 1.

If pR and ε > 0, then the ε-ball in R centered at p is simply the open interval (pε, p + ε).

This real line has several important properties as a metric space:

As a topological space

The real line can be compactified by adding a point at infinity. Real projective line.svg
The real line can be compactified by adding a point at infinity.

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 and metric topology on R are the same. As a topological space, the real line is homeomorphic to the open interval (0, 1).

The real line is trivially a topological manifold of dimension 1. Up to homeomorphism, it is one of only two different connected 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 a locally compact space and a paracompact space, 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 groups 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 R 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.

As a vector space

The bijection between points on the real line and vectors Bijection between vectors and points on number line.svg
The bijection between points on the real line and vectors

The real line is a vector space over the field R of real numbers (that is, over itself) of dimension 1. It has the usual multiplication as an inner product, making it a Euclidean vector space. The norm defined by this inner product is simply the absolute value.

As a measure space

The real line carries a canonical measure, namely the Lebesgue measure. This measure can be defined as the completion of a Borel measure defined on R, 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.

In real algebras

The real line is a one-dimensional subspace of a real algebra A where RA.[ clarification needed ] For example, in the complex plane z = x + iy, the subspace {z : y = 0} is a real line. Similarly, the algebra of quaternions

q = w + x i + y j + z k

has a real line in the subspace {q : x = y = z = 0 }.

When the real algebra is a direct sum then a conjugation on A is introduced by the mapping of subspace V. In this way the real line consists of the fixed points of the conjugation.

See also

Related Research Articles

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Compact space Topological notions of all points being "close"

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Euclidean space Fundamental space of geometry

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Surface (topology) Two-dimensional manifold

In the part of mathematics referred to as topology, a surface is a two-dimensional manifold. Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions of two variables; see the figure at right. However, surfaces can also be defined abstractly, without reference to any ambient space. For example, the Klein bottle is a surface that cannot be embedded in three-dimensional Euclidean space.

Open set Basic subset of a topological space

In mathematics, open sets are a generalization of open intervals in the real line. In a metric space—that is, when a distance is defined—open sets are the sets that, with every point P, contain all points that are sufficiently near to P.

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.

In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a discontinuous sequence, meaning they are isolated from each other in a certain sense. The discrete topology is the finest topology that can be given on a set. Every subset is open in the discrete topology so that in particular, every singleton subset is an open set in the discrete topology.

General topology Branch of topology

In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geometric topology, and algebraic topology. Another name for general topology is point-set 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.

In topology, the long line is a topological space somewhat similar to the real line, but in a certain way "longer". It behaves locally just like the real line, but has different large-scale properties. Therefore, it serves as one of the basic counterexamples of topology. Intuitively, the usual real-number line consists of a countable number of line segments [0, 1) laid end-to-end, whereas the long line is constructed from an uncountable number of such segments.

Real coordinate space Space formed by the n-tuples of real numbers

In mathematics, a real coordinate space of dimension n, written Rn or n, is a coordinate space over the real numbers. This means that it is the set of the n-tuples of real numbers. With component-wise addition and scalar multiplication, it is a real vector space.

Manifold Topological space that locally resembles Euclidean space

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 neighborhood that is homeomorphic to the Euclidean space of dimension n.

In topology, a branch of mathematics, a topological manifold is a topological space which locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topological spaces, with applications throughout mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold. Every manifold has an "underlying" topological manifold, obtained by simply "forgetting" the added structure.

Space (mathematics) Mathematical set with some added structure

In mathematics, a space is a set with some added structure.

Real number Number representing a continuous quantity

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line. The adjective real in this context was introduced in the 17th century by René Descartes, who distinguished between real and imaginary roots of polynomials. The real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as 2. Included within the irrationals are the real transcendental numbers, such as π (3.14159265...). In addition to measuring distance, real numbers can be used to measure quantities such as time, mass, energy, velocity, and many more. The set of real numbers is denoted using the symbol R or and is sometimes called "the reals".

In topology and related areas of mathematics, a subset A of a topological space X is called dense if every point x in X either belongs to A or is a limit point of A; that is, the closure of A constitutes the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, the rational numbers are a dense subset of the real numbers because every real number either is a rational number or has a rational number arbitrarily close to it.

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