Successors of locally compact topological group topologies on abelian groups
Tom 249 / 2020
Streszczenie
For a group , 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.
