Homotopy equivalent but not deformation retraction

Question

Can somebody come up with an example where $X \subset Y$, the inclusion gives a homotopy equivalence between $X$ and $Y$, but there is no deformation retraction from $Y$ onto $X$?

Answer 1

There are known examples of spaces which are contractible but do not deformation retract onto any point in them. So you can take $Y$ to be such a space and $X$ to be any point in it.

Answer 2

Consider the space $$Y=\{(x,y) \in \mathbb R^2 \mid \exists m \in \mathbb Q\ y=mx \}$$ this space is contractible, so it is homotopy equivalent to the point $\{(1,0)\}=X$ for instance, nevertheless $\{(1,0)\}$ is not a deformation retract of $Y$.