I want to find textbook which contains the next proposition(?).
I think it is true, but I can't find proof of that.
Please teach me a textbook in which the next proposition(?) are proved.
Proposition(?) (Characterization of discrete subgroup)
Let $G$ be a topological group and e be identity element of G.
For a subgroup $H \subset G\ $, The following are equivalent.
(1) Relative topology of H is discrete.
(2) There exists an open neighborhood $U$ of $\ e\ \ $such that$\ \ H \bigcap U = \left\{ 0 \right\}$.
(3) There exists an open neighborhood $V$ of $\ e \ \ $such that$\ \ \ \forall x \in G ,\ \#(H \bigcap xV) \leq 1\ $ where$\ \#\ $means cardinality of a set.
(4) When we give the discrete topology to H,$\ \ $action of $H$ to $G\ $:$\ H \times G \rightarrow G$,$\ \ h \mapsto h \cdot g\ $ is properly discontinuous.
Here are some Hints for the exercise.
For (1) and (2)- notice the $e$ is open in the relative topology so $e= H\cap U$
For (2) and (3) - given $V$ as in (3) take $U=V$ and given $U$ as in (2) take $V$ to be a symmetric open set s.t $VV\subset U$.
For (4): (Notice that the term "properly discontinuous" usually assumes $G$ is locally compact. ) If $H$ is discrete then the intersection of $H$ with a compact set is finite, this implies the action is "properly discontinuous" since for any compact set $K\subset G$ then $KK^{-1}$ is compact as well. so #$\{h\ : \ hK\cap K\}=$#$\{h\ : \ h\in KK^{-1} \}<\infty$. In the other direction, if we assume $H$ is not disreate then there is a accumulation point $x$- thus we will not be able to find a $V$ satisfying (3)