(1) General case: Let $X$ be a topological space and $A$ be its subspace. Suppose $X$ and $A$ are homotopically equivalent to each other, then is $A$ necessarily a deformation retract of $X$?
(2) Special case: I think even if the inclusion $i: A \hookrightarrow X$ is a homotopic equivalence, we still don't know whether its homotopic inverse is a retraction $r:X \to A$. In this special case, is $A$ necessarily a deformation retract of $X$?
I am very confused about both (1) and (2) and find it hard to prove them or come up with counterexamples. Thanks in advance!
Corollary 0.20 of Hatcher's book: If $(X, A)$ satisfies the homotopy extension property(CW pair, for instance) and the inclusion $A֓ \hookrightarrow X$ is a homotopy equivalence, then $A$ is a deformation retract of $X$ .