- #1

latentcorpse

- 1,444

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is:

(i) [itex]GL_2(\mathbb{R}) \subset \mathbb{R}^4[/itex] an open subspace?

(ii) [itex]GL_2(\mathbb{R})[/itex] compact?

(iii) [itex]GL_2(\mathbb{R})[/itex] connected?

i'm inclined to say no for (i) but i can't really explain it. if you take an invertible matrix and give it a small perturbation (i.e. change one of the entries by a small amount) it will not necessarily be invertible any more...i.e. it could now be outside the subspace, meaning it's a closed subspace. but i don't know how to put that into maths

for compactness, it needs to be closed and bounded, surely if my answer to (i) is correct then it cannot be compact

finally, i am not sure how to go about (iii) - any advice?