Wikipedia says about deformation retraction that it "…is a special case of homotopy equivalence…"
I fail to see how this is true. Say $A \subset X$ and $F$ a deformation retraction from $id_X$ to $id_A$.
If $F$ was a homotopy equivalence we would have to have $F_0 \circ F_1 \simeq id_X$ and $F_1 \circ F_0 \simeq id_A$.
The first is impossible because $F_1(X) \subset A$.
What am I missing? Thanks for your help.
Best Answer
Even if $Im(id_X) \subset A$, $F_0 \circ F_1 \simeq id_X$ can still be true. Note that $\simeq$ is not $=$.