**Euclidean space**is the usual

*n*-dimensional mathematical space, a generalization of the 2- and 3-dimensional spaces studied by Euclid. Formally, for any non-negative integer

*n*,

*n*-dimensional 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 "*n*-dimensional Euclidean space" is usually abbreviated to "Euclidean *n*-space", or even just "*n*-space". Euclidean *n*-space 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 *n*-manifold, and is in fact a differentiable *n*-manifold. For *n* ≠ 4, any differentiable *n*-manifold 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) 4-spaces.

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 *n*-space can also be considered as an *n*-dimensional 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

**x**·**y**=*x*_{1}*y*_{1}+ ... +*x*_{n}*y*_{n}.