Every LCA group has an open compactly generated subgroup with no open subgroup of infinite index

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

Classification of compactly generated abelian groups. Every compactly generated locally compact Hausdorff abelian group is isomorphic as a topological group to ${\mathbb{R}}^m\times {\mathbb{Z}}^n\times K$ for some $m,n\ge 0$ and some compact group $K$. This is Hewitt-Ross, Abstract Harmonic Analysis I, Theorem 9.8, quoted and attributed in the Ross article recorded in the sources (Theorem 3 proof, p. 3); the primary volume is not available here, so the classification is used as a cited theorem and no minimality claim about its axiom basis is made beyond the declared AC and DC.

An open subgroup is closed because its complement is a union of open cosets; a closed subgroup of an LCA group is LCA by A locally compact subgroup of a Hausdorff topological group is closed. A compactly generated group is one that contains a compact set generating it as a group; the subgroup $⟨C⟩$ generated by a symmetric set $C$ containing the identity is the union of the sets $C^k$ of sums of $k$ elements of $C$, and it is open as soon as $C$ is a neighbourhood of the identity. A locally compact space has compact neighbourhoods, and sums and finite products of compact sets are compact (A product of finitely many compact spaces is compact in the product topology, A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism); closed Euclidean balls are compact (Heine-Borel in ${\mathbb{R}}^n$: with the Euclidean metric a subset of ${\mathbb{R}}^n$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line). (The subgroup $⟨S⟩$ generated by a subset, the cyclic subgroup $⟨g⟩$, and cyclic groups, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open, Locally compact topological space: every point has a compact neighbourhood; and what this says in a metric space, In a locally compact Hausdorff space every open set containing a point contains an open set containing it whose closure is compact and still inside; such a space is regular)

$\mathbb{Z}$ and ${\mathbb{R}}^m$ carry their standard topologies, $\mathbb{Z}$ being discrete and ${\mathbb{R}}^m$ connected, since the real line is order-convex and finite products of connected spaces are connected (A subset of $\mathbb{R}$ is connected if and only if it is order-convex, that is, an interval, A product of connected spaces is connected in the product topology, and that argument is a theorem of ZF; for an infinite index set it is the assertion that the product of nonempty spaces is nonempty that uses the Axiom of Choice); the quotient $G\mathrm{/}L$ of a topological group by a subgroup $L$ carries the quotient topology, which makes the quotient map continuous, and for an open subgroup $L$ every coset is open (translations are homeomorphisms), so $G\mathrm{/}L$ is discrete. (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets, The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$, The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection, Subgroup, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological, Left and right translations and inversion in a topological group are homeomorphisms)

Continuous images of compact sets are compact, and the continuous image of a connected space is connected (A continuous image of a connected space is connected, and connectedness is a topological property). The discrete topology on a finite set is compact while the discrete topology on an infinite set is not. A group homomorphism out of a product that is independent of one factor descends to the quotient by the kernel of that factor (For a quotient map $q:X\to Y$, a map out of $Y$ is continuous iff its composite with $q$ is; a continuous map on $X$ constant on the fibres of $q$ factors uniquely through $q$; and a composite of quotient maps is a quotient map, 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). (A continuous image of a compact space is compact; a continuous real-valued map on a nonempty compact space attains a maximum and a minimum; and a continuous bijection from a compact space to a Hausdorff space is a homeomorphism, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets)