= Mapping-cylinder bottom retraction criterion
{title2=$X\times\{0\}\text{ retracts from }M(f)\iff gf\simeq\operatorname{id}_X$}
Use the <mapping cylinder> convention $(x,1)\sim f(x)$ for $f:X\to Y$. Its bottom copy of $X$ is a <retract> precisely when $f$ has a left <homotopy inverse> $g:Y\to X$. A <retraction> restricted to $Y$ supplies $g$, and its values along the cylinder supply the <homotopy> from $\operatorname{id}_X$ to $gf$. Conversely use that <homotopy> on the cylinder and $g$ on $Y$; agreement at the attached end makes the map descend by the <universal property of the quotient topology>.
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