Euclidean topology

Topological structure of Euclidean space

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

## Definition

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The Euclidean norm on  R n {\displaystyle \mathbb {R} ^{n}} is the non-negative function  ‖ ⋅ ‖ : R n -> R {\displaystyle \|\cdot \|:\mathbb {R} ^{n}\to \mathbb {R} } defined by  ‖ ( p 1 , … , p n ) ‖ := p 1 2 + ⋯ + p n 2 . {\displaystyle \left\|\left(p_{1},\ldots ,p_{n}\right)\right\|~:=~{\sqrt {p_{1}^{2}+\cdots +p_{n}^{2}}}.}

Like all norms, it induces a canonical metric defined by  d ( p , q ) = ‖ p − q ‖ . {\displaystyle d(p,q)=\|p-q\|.} The metric  d : R n × R n -> R {\displaystyle d:\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} } induced by the Euclidean norm is called the Euclidean metric or the Euclidean distance and the distance between points  p = ( p 1 , … , p n ) {\displaystyle p=\left(p_{1},\ldots ,p_{n}\right)} and  q = ( q 1 , … , q n ) {\displaystyle q=\left(q_{1},\ldots ,q_{n}\right)} is  d ( p , q ) = ‖ p − q ‖ = ( p 1 − q 1 ) 2 + ( p 2 − q 2 ) 2 + ⋯ + ( p i − q i ) 2 + ⋯ + ( p n − q n ) 2 . {\displaystyle d(p,q)~=~\|p-q\|~=~{\sqrt {\left(p_{1}-q_{1}\right)^{2}+\left(p_{2}-q_{2}\right)^{2}+\cdots +\left(p_{i}-q_{i}\right)^{2}+\cdots +\left(p_{n}-q_{n}\right)^{2}}}.}

In any metric space, the open balls form a base for a topology on that space.[1] The Euclidean topology on  R n {\displaystyle \mathbb {R} ^{n}} is the topology generated by these balls. In other words, the open sets of the Euclidean topology on  R n {\displaystyle \mathbb {R} ^{n}} are given by (arbitrary) unions of the open balls  B r ( p ) {\displaystyle B_{r}(p)} defined as  B r ( p ) := { x ∈ R n : d ( p , x ) < r } , {\displaystyle B_{r}(p):=\left\\{x\in \mathbb {R} ^{n}:d(p,x)<r\right\\},} for all real  r > 0 {\displaystyle r>0} and all  p ∈ R n , {\displaystyle p\in \mathbb {R} ^{n},} where  d {\displaystyle d} is the Euclidean metric.

## Properties

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When endowed with this topology, the real line  R {\displaystyle \mathbb {R} } is a T5 space. Given two subsets say  A {\displaystyle A} and  B {\displaystyle B} of  R {\displaystyle \mathbb {R} } with  A ¯ ∩ B = A ∩ B ¯ = ∅ , {\displaystyle {\overline {A}}\cap B=A\cap {\overline {B}}=\varnothing ,} where  A ¯ {\displaystyle {\overline {A}}} denotes the closure of  A , {\displaystyle A,} there exist open sets  S A {\displaystyle S_{A}} and  S B {\displaystyle S_{B}} with  A ⊆ S A {\displaystyle A\subseteq S_{A}} and  B ⊆ S B {\displaystyle B\subseteq S_{B}} such that  S A ∩ S B = ∅ . {\displaystyle S_{A}\cap S_{B}=\varnothing .} [2]

## See also

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* Hilbert space – Type of vector space in math
* List of Banach spaces
* List of topologies – List of concrete topologies and topological spaces

## References

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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