Nondiscrete P-Groups Can be Reflexive

TL;DR

This paper constructs examples of nondiscrete reflexive P-groups and shows that products of feathered Abelian groups with P-modified topology are reflexive, answering longstanding questions in topological group theory.

Contribution

It provides the first examples of nondiscrete reflexive P-groups and proves that products of feathered Abelian groups with P-modified topology are reflexive, solving open problems.

Findings

Constructed nondiscrete reflexive P-groups.

Proved products of feathered Abelian groups with P-topology are reflexive.

Answered open questions in the theory of topological groups.

Abstract

We present a series of examples of nondiscrete reflexive P-groups (i.e., groups in which all $G_\delta$-sets are open) as well as noncompact reflexive $\omega$-bounded groups (in which the closure of every countable set is compact). Our main result implies that every product of feathered (equivalently, almost metrizable) Abelian groups equipped with the P-modified topology is a reflexive group. In particular, every compact Abelian group with the P-modified topology is reflexive. This answers a question posed by S. Hern\'andez and P. Nickolas and solves a problem raised by Ardanza-Trevijano, Chasco, Dom\'{\i}nguez, and Tkachenko.