In mathematics, the real line is simply the set of real numbers. However, this term is usually used when R is to be treated as a space of some sort, such as a topological space or a vector space. The real line has been studied at least since the days of the ancient Greeks, but it was not rigorously defined until 1872. Before and since that date, it has been a prolific example that has played a significant role in many branches of mathematics.
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 can be turned into a metric space by using the metric given by the absolute value: d(x,y): = | y - x | . This metric induces a topology on R equal to the order topology.
As a topological space, the real line is a topological manifold of dimension 1. It is paracompact and second-countable as well as contractible and locally compact. 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.) Indeed, R was historically the first example to be studied of each of these mathematical structures, so that it serves as the inspiration for these branches of modern mathematics. (Indeed, many of the terms above can't even be defined until R is already in place.)
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 an Euclidean space. (The inner product is simply ordinary multiplication of real numbers.) As a vector space, it is not very interesting, and thus it was in fact 2-dimensional Euclidean space that was first studied as a vector space. However, we can still say that R inspired the field of linear algebra, since vector spaces were first studied over R.
R is also a premier example of a ring, even a field. It is in fact a real complete field, and was the first such field to be studied, so that it inspired that branch of abstract algebra as well. However, in such purely algebraic contexts, R is rarely called a "line".
For more information on R in all of its guises, see Real number.