Prove Homeomorphism of {A in GL(n;R) | det(A)>0 & det(A)<0}

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Discussion Overview

The discussion revolves around the question of proving that the set of matrices in GL(n;R) with positive determinants is homeomorphic to the set of matrices in GL(n;R) with negative determinants. The scope includes theoretical aspects of topology and properties of homeomorphisms.

Discussion Character

  • Exploratory
  • Technical explanation

Main Points Raised

  • One participant asks for help in proving that the two sets are homeomorphic.
  • Another participant suggests writing down an explicit homeomorphism as a potential approach.
  • A different participant expresses a desire for clarification on how to explain the properties of a homeomorphism in this context.
  • One comment emphasizes considering the topology used in GL(n;R) and the properties of the determinant function to gain clarity on the problem.

Areas of Agreement / Disagreement

Participants appear to be exploring the question without reaching a consensus. There are multiple viewpoints on how to approach the proof, and no definitive resolution is presented.

Contextual Notes

The discussion does not specify the topology being used or the properties of the determinant function in detail, which may limit the clarity of the arguments presented.

rifat
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Could anyone help me to prove the following question:
"How can we show that the set {A in GL(n;R) | det(A)>0} is homeomorphic to the set {A in GL(n;R) | det(A)<0}?"
 
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Have you tried to write down a homeomorphism?
 
Thanks for reply. Yah I mean Homeomorphism. Actually I would ike to know how can we show that the set {A in GL(n;R) | det(A)>0} is homeomorphic to the set {A in GL(n;R) | det(A)<0}. I didnt understand how can we explain the properties of Homeomorphism function here. Could you please tell me a little detail. Thanks again.
 
I meant, have you tried to give an explicit homeomorphism between those two sets? It shouldn't be too hard.
 
Rifat ,I hope this is not too obvious of a comment:
Think of the topology you are using in GL(n;R) , the properties of det (A),and it
should become clearer.
 

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