Euclidean Space

In mathematics, Euclidean space is the Euclidean plane, the three-dimensional space of Euclidean geometry, and generalizations of these ideas to higher dimensions. The term “Euclidean” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity. It is named for the Greek mathematician Euclid of Alexandria.

Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. This approach brings the tools of algebra and calculus to bear on questions of geometry, and has the advantage that it generalizes easily to Euclidean spaces of more than three dimensions.

From the modern viewpoint, there is essentially only one Euclidean space of each dimension. In dimension one this is the real line; in dimension two it is the Cartesian plane; and in higher dimensions it is a coordinate space with three or more real number coordinates – in short, an n-dimensional real coordinate space. A point in Euclidean space may be identified by a tuple of real numbers, and distances are defined using the Euclidean distance formula. Mathematicians often denote the n-dimensional Euclidean space by, or sometimes if they wish to emphasize its Euclidean nature. Euclidean spaces have finite dimension.

Read more about Euclidean SpaceIntuitive Overview, Real Coordinate Space, Euclidean Structure, Topology of Euclidean Space, Generalizations

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Stable Normal Bundle - Construction Via Embeddings - Details
... Given a manifold or other suitable space X, with two embeddings into Euclidean space these will not in general be isotopic, or even maps into the same space ( need not equal ) ... However, one can embed these into a larger space by letting the last coordinates be 0 This process of adjoining trivial copies of Euclidean space is called stabilization ... One can thus arrange for any two embeddings into Euclidean space to map into the same Euclidean space (taking ), and, further, if is sufficiently large, these embeddings are isotopic, which ...
... In mathematics, a manifold of dimension n is a topological space that near each point resembles n-dimensional Euclidean space ... of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n ... near each point, a manifold resembles Euclidean space, globally a manifold might not ...
Euclidean Space - Generalizations
... In modern mathematics, Euclidean spaces form the prototypes for other, more complicated geometric objects ... smooth manifold is a Hausdorff topological space that is locally diffeomorphic to Euclidean space ... Diffeomorphism does not respect distance and angle, so these key concepts of Euclidean geometry are lost on a smooth manifold ...
Locally Connected Space - Background
... Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the ... However, whereas the structure of compact subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, connected subsets of (for n > 1) proved to be much more ... Indeed, while any compact Hausdorff space is locally compact, a connected space – and even a connected subset of the Euclidean plane – need not be locally connected (see below) ...
Stable Normal Bundle - Construction Via Embeddings
... Given an embedding of a manifold in Euclidean space (provided by the theorem of Whitney), it has a normal bundle ... is not unique, but for high dimension of the Euclidean space it is unique up to isotopy, thus the (class of the) bundle is unique, and called the stable ... This construction works for any Poincaré space X a finite CW-complex admits a stably unique (up to homotopy) embedding in Euclidean space, via general position, and this embedding yields a ...

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