Deformation retract and homotopy equivalence If $A\subset X$ is a deformation retract of $X$. Are $X$ and $A$ homotopy equivalent?

If yours is the standard definition, the answer is "yes".
Namely, $A \subset X$ is a deformation retract of $X$ if there is a continuous map $r: X \longrightarrow A$ such that $r\circ i = \mathrm{id}_A$, where $i: A \longrightarrow X$ is the inclusion, and $i\circ r \simeq \mathrm{id}_X$.
Since $=$ implies $\simeq$ and being of the same homotopy type means just that there are maps $f: A \longrightarrow X$ and $g: X \longrightarrow A$ such that $f \circ g \simeq \mathrm{id}_X$ and $g\circ f \simeq \mathrm{id}_A$, obviously being a deformation retract implies being homotopy equivalent, or having the same homotopy type.
