Euclidean space

Definition

Euclidean space of dimension $n$, denoted $\mathbb {R} ^{n}$, is the Cartesian $n^{th}$ power of $\mathbb {R}$, 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.

Relation with other properties

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: $\mathbb {R} ^{n}$ has a basis of open sets, each of which is homeomorphic to the whole of $\mathbb {R} ^{n}$. 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.