Definition 11.2 (Homotopy equivalence):

(i) A map $f : X \longrightarrow Y$ is said to be a homotopy equivalence if there exists a map $g: Y \longrightarrow X$ such that $f\circ g$ and $g\circ f$ are respectively homotopic to the identity maps id$_Y$ and id$_X$ respectively. Under this circumstance we say that the spaces $X$ and $Y$ are homotopically equivalent or have the same homotopy type.

(ii) A space that is homotopy equivalent to a point is said to be contractible. This is equivalent to the statement that the identity map on $X$ is homotopic to a constant map.

The student may check that if $X$ and $Y$ are homotopy equivalent and $Y$ and $Z$ are homotopically equivalent then $X$ and $Z$ are homotopy equivalent.