As a linear continuum 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 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.
As a metric space on the real line is
absolute difference. around a number The real line forms a
metric space, with the
distance function given by absolute difference: : d(x, y) = |x - y|. The
metric tensor is clearly the 1-dimensional
Euclidean metric. Since the -dimensional Euclidean metric can be represented in matrix form as the -by- identity matrix, the metric on the real line is simply the 1-by-1 identity matrix, i.e. 1. 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 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.
As a topological space 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 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 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 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 number 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 of real numbers (that is, over itself) of
dimension . 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 , 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 When
A is a unital
real algebra, the products of real numbers with 1 is a real line within the algebra. For example, in the
complex plane z =
x + i
y, 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 A = R \oplus V, then a
conjugation on
A is introduced by the mapping v \to -v of subspace
V. In this way the real line consists of the
fixed points of the conjugation. For a dimension
n, the
square matrices form a
ring that has a real line in the form of real products with the
identity matrix in the ring. ==See also==