Connected components of upper triangular matrices

[Pietjuh]

Hello, I'm working on a problem in topology. I'm supposed to find the number of connected components of the group of 2x2 invertible upper triangular matrices over R which i shall call $B_2$.

I've tried it a bit, but I don't know for sure if my approach (and answer) is correct.

Since any homeomorphism preserves connectivity, I consider the trivial homeomorphism of $B_2$ to $\mathbf{R}^3$.

Since the matrices have to be invertible the determinant is non-zero which means that for matrices (a b | 0 c), ac > 0 or ac < 0. But he piece with positive determinant splits in 2 non connected pieces, {(a,b,c) | a > 0 and c > 0} and {(a,b,c) | a < 0 and c < 0}. The same sort of thing holds for the piece with negative determinant.

Is it correct to assert from this that $B_2$ has 4 connected components?

 

[matt grime]

I think that shows it splits into at least 4 connected components. You've not show that each of those individual pieces is connected.