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In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces ...
\R^n by the Euclidean metric.


Definition

The Euclidean norm on \R^n is the non-negative function \, \cdot\, : \R^n \to \R defined by \left\, \left(p_1, \ldots, p_n\right)\right\, ~:=~ \sqrt. Like all norms, it induces a canonical metric defined by d(p, q) = \, p - q\, . The metric d : \R^n \times \R^n \to \R induced by the Euclidean norm is called the Euclidean metric or the Euclidean distance and the distance between points p = \left(p_1, \ldots, p_n\right) and q = \left(q_1, \ldots, q_n\right) is d(p, q) ~=~ \, p - q\, ~=~ \sqrt. In any
metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...
, the open balls form a base for a topology on that space. Metric space#Open and closed sets.2C topology and convergence The Euclidean topology on \R^n is the topology by these balls. In other words, the open sets of the Euclidean topology on \R^n are given by (arbitrary) unions of the open balls B_r(p) defined as B_r(p) := \left\, for all real r > 0 and all p \in \R^n, where d is the Euclidean metric.


Properties

When endowed with this topology, the real line \R is a T5 space. Given two subsets say A and B of \R with \overline \cap B = A \cap \overline = \varnothing, where \overline denotes the closure of A, there exist open sets S_A and S_B with A \subseteq S_A and B \subseteq S_B such that S_A \cap S_B = \varnothing.


See also

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References

{{reflist Topology Euclid