Are GL(n;ℝ) matrices with positive and negative determinants homeomorphic?

[rifat]

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}?"

 

[morphism]

Have you tried to write down a homeomorphism?

 

[rifat]

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.

 

[morphism]

I meant, have you tried to give an explicit homeomorphism between those two sets? It shouldn't be too hard.

 

[WWGD]

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.