Euclidean topology

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

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

Properties

• The real line, with this topology, is a T₅ space. Given two subsets, say A and B, of R with A ∩ B = A ∩ B = ∅, where A denotes the closure of A, there exist open sets S_A and S_B with A ⊆ S_A and B ⊆ S_B such that S_A ∩ S_B = ∅.