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Ali Taghavi
Ali Taghavi
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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?

But can such examples be constructed in an algebraic manner? InIn 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?

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 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?

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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Ali Taghavi
Ali Taghavi
  • 356
  • 8
  • 31
  • 123

Two open contractible subset of $\mathbb{R}^3$ which are not homeomorphic but are polynomial preimage of open sets of $\mathbb{R]$R}$

Source Link
Ali Taghavi
Ali Taghavi
  • 356
  • 8
  • 31
  • 123

Two open contractible subset of $\mathbb{R}^3$ which are not homeomorphic but are polynomial preimage of open sets of $\mathbb{R]$

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 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?