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.[1] 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 \( {\displaystyle B_{r}(p)} \) defined as \( {\displaystyle B_{r}(p):=\{x\in \mathbb {R} ^{n}\mid d(p,x)<r\}} \) , 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 SA and SB with A ⊆ SA and B ⊆ SB such that SA ∩ SB = ∅.[2]
References
Metric space#Open and closed sets.2C topology and convergence
Steen, L. A.; Seebach, J. A. (1995), Counterexamples in Topology, Dover, ISBN 0-486-68735-X
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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