Does homotopy equivalence to a subspace imply (weak) deformation retract? Here is a nearly same question, but with a slight difference in interpretation. We use the definition of wikipedia, which is different from that of Hatcher or Munkres (where deformation retract refers to strong deformation retract on wikipedia).
My question is :

Suppose $X$ is a path-connected topological space, and $A$ is its subspace. If $X$ and $A$ are homotopic equivalent, is it true that $A$ is a (weak) deformation retract of $X$?

Both Hatcher and Munkres have provided counterexamples for the case of strong deformation retract, but not the one above. Please help.

There is actually an easy example. The subspace doesn't even need to be a retract: Let $\mathbb{D}$ be the open disk centered at the origin with radius 1 and $\overline{\mathbb{D}}$ its closure. Let X = $\overline{\mathbb{D}}$ our topological space. There cannot exist a continuous function $$ f: \overline{\mathbb{D}} \rightarrow \mathbb{D}$$
since $\mathbb{D}$ is dense in $\overline{\mathbb{D}}$, hence $\mathbb{D}$ is not a retract.
However, both spaces are contractible thus homotopy equivalent.
