Editing Number line (section)

===As a topological space===
[[Image:Real Projective Line (RP1).png|thumb|The real line can be [[Compactification (mathematics)|compactified]] by adding a  [[point at infinity]].]]

The real line carries a standard [[topological space|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 {{math|'''R'''}} are the same.  As a topological space, the real line is [[homeomorphic]] to the open interval {{math|(0, 1)}}.

The real line is trivially a [[topological manifold]] of [[dimension]] {{Num|1}}.  Up to homeomorphism, it is one of only two different connected 1-manifolds without [[manifold with boundary|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 space|normal]].  It is also [[path-connected]], and is therefore [[connected space|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 {{math|'''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 [[End (topology)|ends]], and the resulting end compactification is the [[extended real number line]] {{math|[−∞, +∞]}}.  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]].