User:Dfeuer/Definition:Usual Topology
Definition
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.
User:Dfeuer/Definition:Usual Topology/Rn
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.
User:Dfeuer/Definition:Usual Topology/Power of Reals
User:Dfeuer/Definition:Usual Topology/Power of Reals