homotopy equivalence

Definition Suppose that $X$ and $Y$ are topological spaces and$f:X\to Y$ is a continuous map.If there exists acontinuous map $g:Y\to X$ such that $f\circ g\simeq i{d}_Y$(i.e. $f\circ g$ is http://planetmath.org/node/1584homotopic to the identitymapping on $Y$),and $g\circ f\simeq i{d}_X$, then$f$ is a homotopy equivalence.This homotopy equivalence is sometimes calledstrong homotopy equivalence to distinguish it fromweak homotopy equivalence.

If there exist a homotopy equivalence between the topologicalspaces $X$ and $Y$, we say that $X$ and $Y$ arehomotopy equivalent, or that$X$ and $Y$ are of the same homotopy type.We then write $X\simeq Y$.

0.0.1 Properties

1. Any homeomorphism $f:X\to Y$ is obviously a homotopy equivalence with$g=f^{-1}$.

2. For topological spaces, homotopy equivalence is anequivalence relation.

3. A topological space $X$ is (by definition) contractible,if $X$ is homotopy equivalent to a point, i.e., $X\simeq \{x_0\}$.