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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?

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    $\begingroup$ U and V then should be open intervals, wlog (0,1) no ? $\endgroup$
    – Pietro Majer
    Commented Sep 1 at 5:22
  • $\begingroup$ @PietroMajer Yes we may assume $U=V=(0,1)$ $\endgroup$
    – Ali Taghavi
    Commented Sep 1 at 10:10
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    $\begingroup$ @PietroMajer BTW no because bounded intervals are not algebraic equivalent to unbounded intervals for example $(0,1)$ is not algebraic equivalent to some thing $(0,\infty)$. So I prefer the original formulation: arbitrary open sets $U,V$ $\endgroup$
    – Ali Taghavi
    Commented Sep 1 at 10:36

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