Euclidean topology

In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on Euclidean n-space Rⁿ 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 Rⁿ is then simply the topology generated by these balls. In other words, the open sets of the Euclidean topology on Rⁿ are given by arbitrary union of the open balls ${\textstyle B_{r>0}(p)=\{x\in \mathbb {R} ^{n}\mid d(x,p)<r\}}$, for 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 = ∅.^[2]

References

1. Metric space#Open and closed sets.2C topology and convergence
2. Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X