Topological groups with dense compactly generated subgroups

Abstract

<p>A topological group G is: (i) compactly generated if it contains a compact subset algebraically generating G, (ii) -compact if G is a union of countably many compact subsets, (iii) <sub>0</sub>-bounded if arbitrary neighborhood U of the identity element of G has countably many translates xU that cover G, and (iv) finitely generated modulo open sets if for every non-empty open subset U of G there exists a finite set F such that F U algebraically generates G. We prove that: (1) a topological group containing a dense compactly generated subgroup is both <sub>0</sub>-bounded and finitely generated modulo open sets, (2) an almost metrizable topological group has a dense compactly generated subgroup if and only if it is both <sub>0</sub>-bounded and finitely generated modulo open sets, and (3) an almost metrizable topological group is compactly generated if and only if it is -compact and finitely generated modulo open sets.</p>