Homotopy equivalences, homotopy inverses and spaces of the same homotopy type

Let $X,Y$ be topological spaces. A continuous map $f:X\to Y$ is a homotopy equivalence if there is a continuous map $g:Y\to X$ such that

$g\circ f\simeq\operatorname{id}_X\qquad\text{and}\qquad f\circ g\simeq\operatorname{id}_Y$

in the sense of Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints. Such a $g$ is a homotopy inverse of $f$.

The spaces $X$ and $Y$ have the same homotopy type, or are homotopy equivalent, written $X\simeq Y$, when a homotopy equivalence $X\to Y$ exists.

The equations required of an ordinary inverse have been weakened to homotopies. Neither composite need equal the corresponding identity map, and a homotopy equivalence need not be bijective.