Euclidean topology

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

Definition

The Euclidean norm on ${\displaystyle \mathbb {R} ^{n}}$  is the non-negative function ${\displaystyle \|\cdot \|:\mathbb {R} ^{n}\to \mathbb {R} }$  defined by

$${\displaystyle \left\|\left(p_{1},\ldots ,p_{n}\right)\right\|~:=~{\sqrt {p_{1}^{2}+\cdots +p_{n}^{2}}}.}$$

 

Like all norms, it induces a canonical metric defined by ${\displaystyle d(p,q)=\|p-q\|.}$  The metric ${\displaystyle d:\mathbb {R} ^{n}\times \mathbb {R} ^{n}\to \mathbb {R} }$  induced by the Euclidean norm is called the Euclidean metric or the Euclidean distance and the distance between points ${\displaystyle p=\left(p_{1},\ldots ,p_{n}\right)}$  and ${\displaystyle q=\left(q_{1},\ldots ,q_{n}\right)}$  is

$${\displaystyle d(p,q)~=~\|p-q\|~=~{\sqrt {\left(p_{1}-q_{1}\right)^{2}+\left(p_{2}-q_{2}\right)^{2}+\cdots +\left(p_{i}-q_{i}\right)^{2}+\cdots +\left(p_{n}-q_{n}\right)^{2}}}.}$$

 

In any metric space, the open balls form a base for a topology on that space.^[1] The Euclidean topology on ${\displaystyle \mathbb {R} ^{n}}$  is 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 ${\displaystyle B_{r}(p):=\left\{x\in \mathbb {R} ^{n}:d(p,x)<r\right\},}$  for all real ${\displaystyle r>0}$  and all ${\displaystyle p\in \mathbb {R} ^{n},}$  where ${\displaystyle d}$  is the Euclidean metric.

Properties

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

See also

• Hilbert space – Type of topological vector space
• List of Banach spaces
• List of topologies – List of concrete topologies and topological spaces

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