User:Dfeuer/Definition:Usual Topology

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.

Definition 1

Let $n$ be a strictly positive natural number.

Let $\R$ be the set of real numbers.

The usual topology on $\R^n$ is the topology induced by the Euclidean metric on $\R^n$.

Definition 2

Let $n$ be a strictly positive natural number.

Let $\R$ be the set of real numbers.

The usual topology on $\R^n$ is the product topology on the product $\ds \prod_{i \mathop = 1}^n \R$ where each factor is given the User:Dfeuer/Definition:Usual Topology/Real Line.