Yesterday I asked about an example in Chevalley's book "Theory of Lie Groups I". Well, the second example on pages 38 and 39 made me work a lot to understand.
But there are still some facts I don't understand. I'll put them below.
1) First paragraph, page 39. He says that $s_{2}$s_{1}$r$ $\in$ $g_{1}$ implies that $r$ is in the group generated by $g_{1}$ and $g_{2}$. This must be easy to see, but I'm not convinced yet.
2) Thir paragraph. I'm not convinced that $V$ (with this condition) exists, neither why this fact implies $V$ to be mapped in a continuous univalent way by $\theta$.
3) Still third paragraph. Why A (the complement of $V_{1}$\cup$(-e_{0})$$V_{1}$ in $Sp(1)$ is compact?
4) Right after that, what does "run over all compact neighbourhood" mean?
Sorry for ask too many things. You can find the book by clicking on that link above. I would appreciate if you could help me :) Or maybe discuss the example two, if you want.