Corollary

If the group G is generated by a compact subset, then it is a projective limit of groups isomorphic to groups of the form $\mathbf{R}^p\times \mathbf{T}^q\times \mathbf{Z}^r\times A$, where A is a finite group and $p,q,r$ are positive integers.

According to (ii) of prop. 1 of II, p. 246, the corollary follows from prop. 2.