Editing Euler characteristic (section)

===Homotopy invariance===
Homology is a topological invariant, and moreover a [[homotopy invariant]]: Two topological spaces that are [[homotopy equivalent]] have [[group isomorphism|isomorphic]] homology groups. It follows that the Euler characteristic is also a homotopy invariant.

For example, any [[contractible]] space (that is, one homotopy equivalent to a point) has trivial homology, meaning that the 0th Betti number is 1 and the others 0. Therefore, its Euler characteristic is 1. This case includes [[Euclidean space]] <math>\mathbb{R}^n</math> of any dimension, as well as the solid unit ball in any Euclidean space &mdash; the one-dimensional interval, the two-dimensional disk, the three-dimensional ball, etc.

For another example, any convex polyhedron is homeomorphic to the three-dimensional [[ball (mathematics)|ball]], so its surface is homeomorphic (hence homotopy equivalent) to the two-dimensional [[sphere]], which has Euler characteristic&nbsp;2. This explains why the surface of a convex polyhedron has Euler characteristic 2.