Cardinal invariants and convergence properties of locally minimal groups

TL;DR

This paper investigates the properties of locally minimal groups, establishing conditions for metrizability and constructing examples of minimal groups with specific convergence and topological features.

Contribution

It provides new results on the metrizability of locally minimal groups and constructs novel examples of minimal groups with unique convergence properties.

Findings

Locally essential subgroups have equal tightness and weight as the ambient group.

Radial locally minimal subgroups imply the ambient group is metrizable.

Constructed minimal groups with specific convergence and topological properties.

Abstract

If G is a locally essential subgroup of a compact abelian group K, then: (i) t(G)=w(G)=w(K), where t(G) is the tightness of G; (ii) if G is radial, then K must be metrizable; (iii) G contains a super-sequence S converging to 0 such that |S|=w(G)=w(K). Items (i)--(iii) hold when G is a dense locally minimal subgroup of K. We show that locally minimal, locally precompact abelian groups of countable tightness are metrizable. In particular, a minimal abelian group of countable tightness is metrizable. This answers a question of O. Okunev posed in 2007. For every uncountable cardinal kappa, we construct a Frechet-Urysohn minimal group G of character kappa such that the connected component of G is an open normal omega-bounded subgroup (thus, G is locally precompact). We also build a minimal nilpotent group of nilpotency class 2 without non-trivial convergent sequences having an open normal countably compact subgroup.