Retractions and deformation retracts, with a deformation retraction required to fix the retract pointwise

Let $A\subseteq X$ carry the subspace topology, and let $i:A\hookrightarrow X$ be the inclusion (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

A continuous map $r:X\to A$ is a retraction of $X$ onto $A$ if

$r\circ i=\operatorname{id}_A,$

equivalently, if $r(a)=a$ for every $a\in A$. When such an $r$ exists, $A$ is a retract of $X$.

The subspace $A$ is a deformation retract of $X$ if there are a retraction $r:X\to A$ and a homotopy

$H:\operatorname{id}_X\simeq_A i\circ r.$

Thus $H(x,0)=x$, $H(x,1)=i(r(x))$, and $H(a,t)=a$ for all $a\in A$ and $t\in I$ (Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints). The pair $(r,H)$ is a deformation retraction.

Some sources call the pointwise-fixed condition a strong deformation retract. In this library the word deformation retract always includes it.