Euclidean space is the usual ndimensional mathematical space, a generalization of the 2 and 3dimensional spaces studied by Euclid. Formally, for any nonnegative integer n, ndimensional Euclidean space is the set R^{n} (where R is the set of real numbers) together with the distance function obtained by defining the distance between two points (x_{1}, ..., x_{n}) and (y_{1}, ...,y_{n}) to be the square root of Σ (x_{i}y_{i})^{2}, where the sum is over i = 1, ..., n. This distance function is based on the Pythagorean Theorem and is called the Euclidean metric.
The term "ndimensional Euclidean space" is usually abbreviated to "Euclidean nspace", or even just "nspace". Euclidean nspace is denoted by E^{ n}, although R^{n} is also used (with the metric being understood). E^{ 2} is called the Euclidean plane.
By definition, E^{ n} is a metric space, and is therefore also a topological space. It is the prototypical example of an nmanifold, and is in fact a differentiable nmanifold. For n ≠ 4, any differentiable nmanifold that is homeomorphic to E^{ n} is also diffeomorphic to it. The surprising fact that this is not also true for n = 4 was proved by Simon Donaldson in 1982; the counterexamples are called exotic (or fake) 4spaces.
Much could be said about the topology of E^{ n}, but that will have to wait until a later revision of this article. One important result, Brouwer's invariance of domain, is that any subset of E^{ n} which is homeomorphic to an open subset of E^{ n} is itself open. An immediate consequence of this is that E^{ m} is not homeomorphic to E^{ n} if m ≠ n  an intuitively "obvious" result which is nonetheless not easy to prove.
Euclidean nspace can also be considered as an ndimensional real vector space, in fact a Hilbert space, in a natural way. The inner product of x = (x_{1},...,x_{n}) and y = (y_{1},...,y_{n}) is given by
See also: Euclidean geometry.
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