Successors of locally compact topological group topologies on abelian groups
Volume 249 / 2020
Abstract
For a group $G$, let $\mathcal{G} (G)$ be the lattice of all topological group topologies on $G$. We prove that if $G$ is abelian, $\tau ,\sigma \in \mathcal{G} (G)$ and $\sigma $ is a successor of $\tau $ in $\mathcal{G} (G)$, then $\sigma $ is precompact iff $\tau $ is precompact. This fact is used to show that if a divisible or connected topological abelian group $(G,\tau )$ contains a discrete subgroup $N$ such that $G/N$ is compact, then $\tau $ does not have successors in $\mathcal {G}(G)$. In particular, no compact Hausdorff topological group topology on a divisible abelian group $G$ has successors in $\mathcal {G}(G)$ and the usual interval topology on $\mathbb {R}$ has no successors in $\mathcal {G}(\mathbb {R})$.
We also prove that a compact Hausdorff topological group topology $\tau $ on an abelian group $G$ has a successor in $\mathcal{G} (G)$ if and only if there exists a prime number $p$ such that $G/pG$ is infinite. Therefore, the usual compact topological group topology of the group $\mathbb Z _p$ of $p$-adic integers does not have successors in $\mathcal{G} (\mathbb Z _p)$.
Our results solve two problems posed by different authors in the years 2006–2018.
