User:Dfeuer/Definition:Usual Topology/Real Line

Definition 1

The usual topology on the reals is defined as the topology induced on $\R$ by the absolute value metric, which is the same as the Euclidean metric on $\R$.

Definition 2

The usual topology on the reals is defined as the topology generated by the basis consisting of all open intervals in $\R$ with the usual ordering. That is, the topology generated by the basis:

$\BB = \set {\openint a b: a, b \in \R}$

Definition 3

The usual topology on the reals is defined as the order topology on $\R$ with the usual ordering.