About  2 decades ago I heared from some one that there are infinitely many open  contractible  subsets of  space  mutually non homeomorphic  to each other. I  confess that I  do not remember the  details of  construction but I remember the  construction was not  so  routine(If I am not mistaken the  method was based on infinitness of the  set of prime numbers..)

But can  such examples be  constructed in an algebraic and semialgebraic manner? 

 >In particular are there two polynomials  $H,G: \mathbb{R}^3 \to \mathbb{R}$  and  two open sets  $U,V$ in $\mathbb{R}$ such that $H^{-1}(U)$  and  $G^{-1}(V)$ are  non homeomorphic contractible  sets?