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SL(n,R) is a closed set of GL(n,R)

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New member
Oct 15, 2013
19
Hello everyone

How to show that the special linear group $SL(n,$R$)$ is a closed set of the general linear group $GL(n,$R$)$

I try to show its compliment is an open set by taking every elements in it has an open neighborhood.

However, I do not know how to define the open neighborhood in this case.
Hope someone can help me out.

Thanks.

Note: special linear group means all element has determinant 1
General linear group means the determinant not equal to $0$
 

Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,712
How to show that the special linear group $\text{SL}(n,\mathbb{R})$ is a closed set of the general linear group $\text{GL}(n,\mathbb{R})$
Hint: use the result, from one of your other threads, that the determinant is a continuous function.