Homotopy equivalence

Remarks

Every homeomorphism is a homotopy equivalence, with its inverse as homotopy inverse. Example: let $i \colon S^1 \to \R^2 \setminus \set{0}$ be the inclusion and $r(x) = x / \norm{x}$. Then $r \circ i = \id_{S^1}$, and $H(x, t) = (1 - t) x + t x / \norm{x}$ is a homotopy in $\R^2 \setminus \set{0}$ from $\id$ to $i \circ r$, so $i$ is a homotopy equivalence. Non-example: the constant map $S^1 \to \set{p}$ is not a homotopy equivalence, since the fundamental group of $S^1$ is $\Z$ and that of a point is trivial. Composites of homotopy equivalences are homotopy equivalences.