The principal structure theorem for LCA groups

Given: A locally compact Hausdorff abelian group $G$, and the Axiom of Choice together with Dependent Choice.

Every locally compact Hausdorff abelian group contains an open, hence closed, subgroup which is compactly generated and contains no open subgroup of infinite index. (Every LCA group has an open compactly generated subgroup with no open subgroup of infinite index)

A compactly generated locally compact Hausdorff abelian group with no open subgroup of infinite index is isomorphic as a topological group to $W\oplus {\mathbb{R}}^n$ for some compact group $W$ and some $n\ge 0$. (A compactly generated LCA group with no open subgroup of infinite index is Euclidean times compact)

A closed subgroup of an LCA group is LCA (A locally compact subgroup of a Hausdorff topological group is closed). An open subgroup of a topological group is closed, and a topological isomorphism carries open subgroups to open subgroups and compact groups to compact groups; products carry the product topology. (Subgroup, Topological group: multiplication and inversion are continuous, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological, The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right)