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.

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