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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.
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 
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 real line forms a metric space, with the metric given by absolute difference:
- d(x, y) = | x − y | .
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 sequence of points converges.
- The real line is path-connected, and is one of the simplest examples of a geodesic metric space
- The Hausdorff dimension of the real line is equal to one.
- The isometry group of the real line, also known as the Euclidean group E(1), consists of all functions of the form x ↦ t ± x, where t is a real number. This group is isomorphic to a semidirect product of the additive group of R with a cyclic group of order two, and is an example of a generalized dihedral group.
As a topological 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 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 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 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 real line is a vector space over the field R of real numbers (that is, over itself) of dimension 1. 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 R is simply the absolute value function.
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.