Homotopy type

Remarks

The relation $\simeq$ is reflexive, symmetric and transitive, so it behaves as an equivalence relation on topological spaces. Symmetry holds because a homotopy inverse is itself a homotopy equivalence. Example: $\R^n$ has the homotopy type of a point; the maps $\R^n \to \set{0}$ and $\set{0} \to \R^n$ are mutually inverse up to the homotopy $H(x, t) = (1 - t) x$. Non-example: $S^1$ and a point do not have the same homotopy type, since their fundamental groups differ. Homeomorphic spaces have the same homotopy type, but the converse fails, as $\R$ and a point show.