How to call two subsets that can be deformed into each other? Given a topological space $X$, is there a canonical name for the equivalence relation generated by the following relation on the subsets of $X$?
$A \sim B :\Leftrightarrow \exists \text{ continuous } h:[0,1]\times X \to X,\; h(0, •) = id_X,\; h(1, A) = B$
that is $A$ can be “homotopically deformed” to match $B$.

I would add to the previous answer that f $A \subset B$ or $B \subset A$ and the application $h|_A=Id_X$ (assuming that $A \subset B$) then you have what is called a $\textbf{Deformation retract}$ (http://en.wikipedia.org/wiki/Deformation_retract) and the application is called a retraction.
This has some interesting properties. The most important one is that, if $A$ is a deformation retract of $B$ then both $A$ and $B$ have the same fundamental group. A common example of this is to retract the cylinder to a circunference along their generatrices. So, the circunference and the cylinder have the same fundamental group:
$$ \pi(\mathbb{S}^1)=\pi(Cylinder)=\mathbb{Z} $$
P.D. I would have add this as a comment rather than an answer but I can't post comments yet.
