Since Euclidean space is a metric space it is also a topological space with the natural topology induced by the metric.

The metric topology on E < sup > n </ sup > is called the Euclidean topology.

A set is open in the Euclidean topology if and only if it contains an open ball around each of its points.

In topology, an n-sphere is defined as a space homeomorphic to the boundary of an ( n + 1 )- ball ; thus, it is homeomorphic to the Euclidean n-sphere, but perhaps lacking its metric.

For example, the real line is Tychonoff under the standard Euclidean topology.

More generally, Euclidean n-space R < sup > n </ sup > with addition and standard topology is a topological group.

For instance, the general linear group GL ( n, R ) of all invertible n-by-n matrices with real entries can be viewed as a topological group with the topology defined by viewing GL ( n, R ) as a subset of Euclidean space R < sup > n × n </ sup >.

The most valuable results, which were obtained by Kazimierz Kuratowski after the war are those that concern the relationship between topology and analytic functions ( theory ), and also research in the field of cutting Euclidean spaces.

* Invariance of domain, a theorem in topology about homeomorphic subsets of Euclidean space

A global geometry is also called a topology, as a global geometry is a local geometry plus a topology, but this terminology is misleading because a topology does not give a global geometry: for instance, Euclidean 3-space and hyperbolic 3-space have the same topology but different global geometries.

For a flat or a hyperbolic spatial geometry, the topology can be either compact or infinite: for example, Euclidean space is flat and infinite, but the torus is flat and compact.

A hermitian matrix of unit determinant and having positive eigenvalues can be uniquely expressed as the exponential of a traceless hermitian matrix, and therefore the topology of this is that of n < sup > 2 </ sup >- 1 dimensional Euclidean space.

Since the latter matrices can be uniquely expressed as the exponential of symmetric traceless matrices, then this latter topology is that of ( n + 2 )( n-1 )/ 2 dimensional Euclidean space.

The unitary group U ( n ) is endowed with the relative topology as a subset of M < sub > n </ sub >( C ), the set of all n × n complex matrices, which is itself homeomorphic to a 2n < sup > 2 </ sup >- dimensional Euclidean space.

Invariance of domain is a theorem in topology about homeomorphic subsets of Euclidean space R < sup > n </ sup >.

The reverse implications do not hold, for example, standard Euclidean space ( R < sup > n </ sup >) is σ-compact but not compact, and the lower limit topology on the real line is Lindelöf but not σ-compact.

* Gaussian measure on Euclidean space with its Borel topology and sigma algebra ;

metals ( examples of degenerate matter and a Fermi gas ), have a 3-dimensional Euclidean topology.

It turns out that they mostly fall into four infinite families, the " classical Lie algebras " A < sub > n </ sub >, B < sub > n </ sub >, C < sub > n </ sub > and D < sub > n </ sub >, which have simple descriptions in terms of symmetries of Euclidean space.

units to some multiple of z, where z is any Gaussian integer ; this turns Z into a Euclidean domain, where

If X is the Euclidean plane with the usual distance and the Lebesgue measure then this question generalizes the classical isoperimetric problem to planar regions whose boundary is not necessarily smooth, although the answer turns out to be the same.

In order to carry out these studies, Hilbert introduced the concept of an infinite dimensional Euclidean space, later called Hilbert space.

The standard way to do this, as carried out in the remainder of this article, is to define the Euclidean plane as a two-dimensional real vector space equipped with an inner product.

# Give the answer as a fraction representing a rational number, so the result of the division of 26 by 11 is But, usually, the resulting fraction should be simplified: the result of the division of 52 by 22 is also This simplification may be done by factoring out the greatest common divisor computed by mean of Euclidean algorithm.

While Lobachevsky created a non-Euclidean geometry by negating the parallel postulate, Bolyai worked out a geometry where both the Euclidean and the hyperbolic geometry are possible depending on a parameter k. Bolyai ends his work by mentioning that it is not possible to decide through mathematical reasoning alone if the geometry of the physical universe is Euclidean or non-Euclidean ; this is a task for the physical sciences.

On the one hand, affine geometry is Euclidean geometry with congruence left out, and on the other hand affine geometry may be obtained from projective geometry by the designation of a particular line or plane to represent the points at infinity.

Euclidean domains are integral domains in which the Euclidean algorithm can be carried out.

In the same way, we expect that the Ricci flow, applied to a distorted round sphere, will tend to round out the geometry over time, but not to turn it into a flat Euclidean geometry.

The first such solutions were discovered in the case of four-dimensional Euclidean space compactified to the four-dimensional sphere, and turned out to be localized in space-time, prompting the names pseudoparticle and instanton.

In geometrical terms, when restricted to real numbers, it bounds the volume in Euclidean space of n dimensions marked out by n vectors v < sub > i </ sub > for 1 ≤ i ≤ n in terms of the lengths of these vectors || v < sub > i </ sub >||.

The notion of a projective plane arises out of the idea of perspective in geometry and art: that it is sometimes useful to include in the Euclidean plane an additional " imaginary " line that represents the horizon that an artist painting the plane might see.

Poncelet discovered the following theorem in 1822: Euclidean compass and straightedge constructions can be carried out using only a straightedge if a single circle and its center is given.

Whereas in Euclidean geometry moving steadily in an orthogonal direction to a ray from the origin traces out a circle, in a pseudo-Euclidean plane steadily moving orthogonal to a ray from the origin traces out a hyperbola.

For n = 2 this yields a point-line duality pointing out why the mathematical foundations of parallel coordinates are developed in the Projective rather than Euclidean space.

the linking number in Euclidean 3-space ( or in the 3-sphere ) of a < sub > i </ sub > and the pushoff of a < sub > j </ sub > out of the surface, with

These can be carried out in Euclidean space, particularly in dimensions 2 and 3.

showed that dimensional regularization is mathematically well defined, at least in the case of massive Euclidean fields, by using the Bernstein – Sato polynomial to carry out the analytic continuation.

The work of Tarski and his students on Euclidean geometry culminated in the monograph Schwabhäuser, Szmielew, and Tarski ( 1983 ), which set out the 10 axioms and one axiom schema shown below, the associated metamathematics, and a fair bit of the subject.

This construction can be carried out geometrically in the three-dimensional Euclidean space R < sup > 3 </ sup >.

Angles are usually presumed to be in a Euclidean plane, but are also defined in non-Euclidean geometry.

For much more distant objects the Euclidean approximation is not valid, and General Relativity must be taken into account when calculating the luminosity distance of an object.

In the Euclidean approximation for nearby objects, the absolute magnitude of a star can be calculated from its apparent magnitude and parallax:

This system can also be used for three-dimensional geometry, where every point in Euclidean space is represented by an ordered triple of coordinates ( x, y, z ).

These definitions are designed to be consistent with the underlying Euclidean geometry.

A pair of Bézout coefficients ( in fact the ones that are minimal in absolute value ) can be computed by the extended Euclidean algorithm.

It follows that the proof that follows may be adapted for any Euclidean domain.

In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis ( in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutral geometry, i. e. no parallel postulate.

The resulting compactification can be thought of as a circle ( which is compact as a closed and bounded subset of the Euclidean plane ).

A point in the Euclidean plane is a constructible point if, given a fixed coordinate system ( or a fixed line segment of unit length ), the point can be constructed with unruled straightedge and compass.

The convolution can be defined for functions on groups other than Euclidean space.

Graphing calculators can be used to graph functions defined on the real line, or higher dimensional Euclidean space.

These Euclidean transformations of the plane can all be described in a uniform way by using matrices.

To be orthogonal, the matrix A must have orthogonal rows with same Euclidean length of one, that is,

Let x, y, z be a system of Cartesian coordinates in 3-dimensional Euclidean space, and let i, j, k be the corresponding basis of unit vectors.

Since traditional " Euclidean " space never reaches infinity, the projective equivalent, called extended Euclidean space, must be formed by adding the required ' plane at infinity '.

The problem of finding the Delaunay triangulation of a set of points in d-dimensional Euclidean space can be converted to the problem of finding the convex hull of a set of points in ( d + 1 )- dimensional space, by giving each point p an extra coordinate equal to | p |< sup > 2 </ sup >, taking the bottom side of the convex hull, and mapping back to d-dimensional space by deleting the last coordinate.

The Euclidean minimum spanning tree of a set of points is a subset of the Delaunay triangulation of the same points, and this can be exploited to compute it efficiently.

In a linear system the phase space is the N-dimensional Euclidean space, so any point in phase space can be represented by a vector with N numbers.

These can be found by applying the extended Euclidean algorithm.

In this sense, Euclidean geometry is more concrete than many modern axiomatic systems such as set theory, which often assert the existence of objects without saying how to construct them, or even assert the existence of objects that cannot be constructed within the theory.

Because of Euclidean geometry's fundamental status in mathematics, it would be impossible to give more than a representative sampling of applications here.

The fundamental types of measurements in Euclidean geometry are distances and angles, and both of these quantities can be measured directly by a surveyor.