A compactly generated LCA group with no open subgroup of infinite index is Euclidean times compact

Given: A compactly generated locally compact Hausdorff abelian group $H$ containing no open subgroup of infinite index, 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.

$\mathbb{Z}$ is an infinite discrete abelian group and ${\mathbb{Z}}^n$ is its $n$-fold product with the product topology; for $n\ge 1$ the map ${\mathbb{R}}^m\times {\mathbb{Z}}^n\times K\to {\mathbb{Z}}^n$, $(v,z,k)\mapsto z$, is a continuous surjective homomorphism with kernel ${\mathbb{R}}^m\times \{0\}\times K$, so by the universal property of the quotient the quotient group $({\mathbb{R}}^m\times {\mathbb{Z}}^n\times K)\mathrm{/}({\mathbb{R}}^m\times \{0\}\times K)$ is topologically isomorphic to ${\mathbb{Z}}^n$. (The integers as equivalence classes of pairs of naturals, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, 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, 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, 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)

A subgroup $L$ of a topological group is open exactly when each of its cosets is open, since translations are homeomorphisms; in the quotient by an open subgroup all points are open, so the quotient is discrete. The set ${\mathbb{R}}^m\times \{0\}\times K$ is open in ${\mathbb{R}}^m\times {\mathbb{Z}}^n\times K$, because $\{0\}$ is open in the discrete group ${\mathbb{Z}}^n$ and $K$ is open in $K$, and the preimage of an open set under a homeomorphism is open. (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, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies)

The index of a subgroup $L\le H$ is the cardinality of the quotient $H\mathrm{/}L$; an infinite quotient group therefore means infinite index. A topological isomorphism preserves subgroups, openness and indices. (The quotient group $G\mathrm{/}N$ and coset product $(gN)(hN)=ghN$, Subgroup, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological)