# Euclidean space

From Topospaces

## Definition

**Euclidean space** of dimension , denoted , is the Cartesian power of , 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: has a basis of open sets, each of which is homeomorphic to the whole of . 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.