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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.

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    @Theo: thanks! Yes, see you around ; )2011-04-13

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Even if $Im(id_X) \subset A$, $F_0 \circ F_1 \simeq id_X$ can still be true. Note that $\simeq$ is not $=$.