I would like to ask for references about compact elements of topological groups. For the record, an element $g$ of a topological group is compact if $g$ lies in a compact subgroup.
I specifically would like to study the proof for the following results:
Proposition 1. The set of compact elements forms a compact subgroup.
Proposition 2. Considering $G$ a locally compact, connected and abelian group. The set of compact elements are dense in $G$
I don't know precisely if that is the hypothesis we need for these results.
Thank you.