For a locally quasi-convex topological abelian group (

*G*,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (

*G*) that are compatible with (τ) (

*i.e.*, have the same dual as (

*G*,τ) ordered by inclusion.

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For a locally quasi-convex topological abelian group (

*G*,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (

*G*) that are compatible with (τ) (

*i.e.*, have the same dual as (

*G*,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(

*G*,\(\widehat{G})\) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (

*H*) of (

*G*) that are easier to handle and show that \(\mathscr{C} (H)\) and \(\mathscr{C} (G/H)\) are large and embed, as a poset, in \(\mathscr{C}(G,τ)\). Important special results are: (i) if \(K\) is a compact subgroup of a locally quasi-convex group \(G\), then \(\mathscr{C}(G)\) and \(\mathscr{C}(G/K)\) are quasi-isomorphic (3.15); (ii) if (

*D*) is a discrete abelian group of infinite rank, then \(\mathscr{C}(D)\) is quasi-isomorphic to the poset \(\mathfrak{F}_D\) of filters on

*D* (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group \(G \) with an open subgroup of infinite co-rank (this class includes, among others, all non-

*σ*-compact LCA groups), the poset \( \mathscr{C} (G) \) is as big as the underlying topological structure of (

*G*,τ) (and set theory) allows. For a metrizable connected compact group \(X\), the group of null sequences \(G=c_0(X)\) with the topology of uniform convergence is studied. We prove that \(\mathscr{C}(G)\) is quasi-isomorphic to \(\mathscr{P}(\mathbb{R})\) (6.9).

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