Euclidean topology

In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on Euclidean n-space R n {displaystyle mathbb {R} ^{n}} by the Euclidean metric. In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on Euclidean n-space R n {displaystyle mathbb {R} ^{n}} by the Euclidean metric. In any metric space, the open balls form a base for a topology on that space. The Euclidean topology on R n {displaystyle mathbb {R} ^{n}} is then simply the topology generated by these balls. In other words, the open sets of the Euclidean topology on R n {displaystyle mathbb {R} ^{n}} are given by (arbitrary) unions of the open balls B r ( p ) {displaystyle B_{r}(p)} defined as B r ( p ) := { x ∈ R n ∣ d ( p , x ) < r } {displaystyle B_{r}(p):={xin mathbb {R} ^{n}mid d(p,x)<r}} , for all r > 0 {displaystyle r>0} and all p ∈ R n {displaystyle pin mathbb {R} ^{n}} , where d {displaystyle d} is the Euclidean metric.