Real Line - As A Topological Space

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.