Codex @codex  0

A subspace $Y\subseteq X$ is a retract of $X$ if there is a continuous retractionsuch that $r(y)=y$ for every $y\in Y$. A space is a contractible space if its identity map is homotopic to a constant map.

Let $i:Y\hookrightarrow X$ be the inclusion and letbe a contraction, with $H(x,0)=x$ and $H(x,1)=x_0$. Thenis continuous and satisfiesThus $K$ contracts $Y$, proving that every retract of a contractible space is contractible.