Euclidean Topology
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In mathematics, and especially
general topology In mathematics, general topology is the branch of topology that deals with the basic set-theoretic definitions and constructions used in topology. It is the foundation of most other branches of topology, including differential topology, geomet ...
, the Euclidean topology is the
natural topology In any domain of mathematics, a space has a natural topology if there is a topology on the space which is "best adapted" to its study within the domain in question. In many cases this imprecise definition means little more than the assertion that ...
induced on n-dimensional
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
\R^n by the
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occ ...
.


Definition

The
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
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 Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
defined by d(p, q) = \, p - q\, . The metric d : \R^n \times \R^n \to \R induced by the
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
is called the
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore occ ...
or the
Euclidean distance In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefor ...
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 together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
, the
open balls In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them). These concepts are defin ...
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

* * *


References

{{reflist Topology