Retract

A subspace $A\subseteq X$ is a retract of $X$ when there is a continuous map $r:X\to A$ whose restriction to $A$ is the identity.

A deformation retraction of $X$ onto $A$ is a homotopy from ${\mathrm{id}}_X$ to a retraction $r:X\to A$ that keeps $A$ inside itself throughout. It makes the inclusion $A\hookrightarrow X$ a homotopy equivalence.

Every retract of a contractible space is contractible. If $i:A\hookrightarrow X$ is the inclusion, $r:X\to A$ is a retraction, and $H$ contracts $X$, thencontracts $A$.