Let $G$ be a lie group, $\Gamma < G$ a discrete subgroup.
Is it true that the canonical map $p: G\to G/\Gamma$ necessarily proper (that is, pre-images of compact sets are compact)?
I know that it is true for compact $\Gamma$, and could not make the proof work for a discrete subgroup. This is false for a general topological space.
(I really don't know if it's true or not, I just thought it would be useful if it is).
Actually, I would also like to get any recommendations for basic textbooks on Lie Groups / Topological Groups that cover such "basic" topological concepts.
P.S:
I understand that my question is really not interesting in the current form (should have not asked it, too trivial), but let me ask what I really needed (which could also be trivial?):
Given $K\subset G/\Gamma$ compact, is there necessarily $K_1 \subset G$ compact such that $p(K_1)=K$?
That is I don't require the pre-image to be compact, just that I'm able to take a compact set of representatives. As far as I can tell, the counterexample $\mathbb{R}/\mathbb{Z}$ fails here.
No. Just take $G=(\mathbb R,+)$ and $\Gamma=\mathbb Z$. Then $p^{-1}\bigl(0+\mathbb{Z}\bigr)=\mathbb Z$, which is not compact.