finiteness of an abelian topological group

Question

Let A be an abelian (Hausdorff) topological group. Assume that

(1) the set of its torsion elements, and

(2) a finitely generated subgroup

are dense subsets of A.

My question: must A be finite?

(This is clearly true if A is discrete or if the f.g. subgroup is torsion).

EDIT. I'm more interested in compact groups.

Answer 1

Consider $S^1$, the multiplicative group of complex numbers of absolute value $1$. The torsion elements are roots of unity, which are dense in $S^1$. Also the subgroup generated by any non-torsion element, say $\exp(2\pi it)$ with $t$ irrationals, is dense in $S^1$.

Of course, $S^1$ is compact and infinite.