Euclidean space

Definition
Euclidean space of dimension $$n$$, denoted $$\R^n$$, is the Cartesian $$n^{th}$$ power of $$\R$$, endowed with the product topology.

Properties of Euclidean space are very important because they govern the properties of manifolds, which are spaces modelled locally on Euclidean spaces.

Weaker properties

 * Completely metrizable space: This follows from the fact that the reals form a complete topological field
 * Convex metrizable space
 * Contractible space: This follows from the fact that Euclidean space is convex metrizable. It is also the reason for all manifolds and all CW-spaces being locally contractible
 * Self-based space: $$\R^n$$ has a basis of open sets, each of which is homeomorphic to the whole of $$\R^n$$. This is the reason why all locally Euclidean spaces (in particular, all manifolds) are uniformly based
 * Compactly nondegenerate space: Any homotopy at a point can be extended to the whole Euclidean space in such a way that the homotopy has compact support in the sense that it does not move the complement of a compact set at all. This is the reason why the inclusion of a point in a manifold is a cofibration
 * Compactly homogeneous space: Given any two points, there is a self-homeomorphism taking one to the other, which is identity outside a compact set. This is the reason why connected manifolds are homogeneous.