I've come across this question during revision and don't really know what you would say? Any help? Regard a 2 x 2 matrix A as a topological space by considering 2x2 matrices as vectors (a,b,c,d) as a member of R4. Let GL2(R) c R4 be the subset of the 2x2 matrices A which are invertible, i.e. such that ad does not equal bc. Consider the following, giving reasons: (i) Is GL2(R) c R4 an open subspace? (ii) Is GL2(R) compact? (iii) Is GL2(R) connected?