Euclidean topology

In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on Euclidean n-space ${\displaystyle \mathbb {R} ^{n}}$ by the Euclidean metric.

In any metric space, the open balls form a base for a topology on that space.[1] The Euclidean topology on ${\displaystyle \mathbb {R} ^{n}}$ is then simply the topology generated by these balls. In other words, the open sets of the Euclidean topology on ${\displaystyle \mathbb {R} ^{n}}$ are given by (arbitrary) unions of the open balls ${\displaystyle B_{r}(p)}$ defined as ${\displaystyle B_{r}(p):=\{x\in \mathbb {R} ^{n}\mid d(p,x)<r\}}$, for all ${\displaystyle r>0}$ and all ${\displaystyle p\in \mathbb {R} ^{n}}$, where ${\displaystyle d}$ is the Euclidean metric.