User:Dfeuer/Definition:Usual TopologyFrom ProofWiki < User:Dfeuer Jump to navigation Jump to searchDefinitionUser:Dfeuer/Definition:Usual Topology/Real LineDefinition 1The 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 $\R$. Show Definition 2The 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 3The usual topology on the reals is defined as the order topology on $\R$ with the usual ordering. User:Dfeuer/Definition:Usual Topology/RnDefinition 1Let $n$ 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 2Let $n$ 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 RealsUser:Dfeuer/Definition:Usual Topology/Power of Reals User:Dfeuer/Definition:Usual Topology/Complex NumbersUser:Dfeuer/Definition:Usual Topology/Complex Numbers Retrieved from "https://proofwiki.org/w/index.php?title=User:Dfeuer/Definition:Usual_Topology&oldid=140171" |