Locally quasi-convex compatible topologies on locally compact abelian groups

Abstract

Let G be a locally compact abelian group (LCA group). Our aim is to study the poset ${\mathcal {C}}(G)$ of all locally quasi-convex topologies on G that are compatible (i.e., have the same dual as G) ordered by inclusion. This poset has a bottom element, namely the Bohr topology $\sigma (G, {{\widehat{G}}})$ and a top element, namely the original topology. We shall be interested in both quantitative aspects of this poset (its size) and its qualitative aspects (its “width”, namely its anti-chain number). Since we are mostly interested in estimates “from below”, our strategy will be to embed well known posets such as the poset of free filters or the poset $([\omega ]^\omega , \subseteq ^*)$ in ${\mathcal {C}}(G)$. We show that for every LCA group G the set of compatible topologies ${\mathcal {C}}(G)$ is order-isomorphic to ${\mathcal {C}}(D)$ for a suitable discrete group D, e.g. ${\mathcal {C}}({\mathbb {R}}^n)\cong {\mathcal {C}}({\mathbb {Z}}^n)$ for all $n\in {\mathbb {N}}$. If the rank of D is infinite, it was already shown in Außenhofer et al. (Axioms 4:436–458, 2019) that ${\mathcal {C}}(D)$ is as big as possible. In this paper we prove that ${\mathcal {C}}({\mathbb {R}})\cong {\mathcal {C}}({\mathbb {Z}})$ and ${\mathcal {C}}({\mathbb {Z}}(p^\infty ))$ have width and hence size at least ${\mathfrak {c}}$ and that they contain chains of length ${\mathfrak {c}}$. This yields that for any non-compact LCA group G the set ${\mathcal {C}}(G)$ has width $\ge {\mathfrak {c}}$ and chains of length $\ge {\mathfrak {c}}$. Further, we characterize the discrete groups D such that ${\mathcal {C}}(D)$ may fail to have the maximal possible cardinality and width (they are at most countably many, up to isomorphism).