what is the standard base for the euclidean topology on the set of real R
Answers
Step-by-step explanation:
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on Euclidean n-space {\displaystyle \mathbb {R} ^{n}}\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}}\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}}\mathbb {R} ^{n} are given by (arbitrary) unions of the open balls {\displaystyle B_{r}(p)}{\displaystyle B_{r}(p)} defined as {\displaystyle B_{r}(p):=\{x\in \mathbb {R} ^{n}\mid d(p,x)<r\}}{\displaystyle B_{r}(p):=\{x\in \mathbb {R} ^{n}\mid d(p,x)<r\}}, for all {\displaystyle r>0}r > 0 and all {\displaystyle p\in \mathbb {R} ^{n}}p \in \mathbb{R}^n, where {\displaystyle d}d is the Euclidean metric.