Retraction and Homotopy type

Question

I need to describe an example of a subspace $A \subset X$ such that there is a retraction $r: X\rightarrow A$ but such that $A$ and $X$ do not have the same homotopy type? any hints? I was thinking about the torus $T^{2}$ and its subspace $A = S^{1}\times \{1\}$

Answer 1

Your example works. They have distinguished fundamental groups and it's clearly a retraction.

Probably the easiest example is any non-empty space which is not contractible. There is always a retraction of a non-empty space onto a single point which is contractible and so not homotopy equivalent to the full space.