It is a problem I encountered when working on shape optimization(mech eng, not math one).
Consider two connected sets $A$ and $B$ in $\mathbb{R}^d$ (it would be nice if $d$ can be chosen arbitrarily, but I only need the result for $d\leq 3$), and each of them contain more than one point.
I am wondering if there always exists a bijective mapping from $A$ to $B$?
Updated assumption:
Further assume there is no degeneracy(I'm not sure if it's a correct word) between two sets.
An example: if $A$ is a 3-dim object, e.g. a cube, $B$ is also a 3-dim object, e.g. a ball. But $B$ cannot be an ojbect possible to be embedded in lower dimensional spaces, e.g. a flat plate is not allowed.
With this imposed assumption, would it further simplify the argument?