Nondiscrete P-Groups Can be Reflexive

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.

💡 Research Summary

The paper addresses a long‑standing open problem in the theory of topological abelian groups: whether non‑discrete P‑groups can be reflexive. A P‑group is a topological group in which every Gδ‑set is open; traditionally such groups were thought to be either discrete or to have very restrictive structure, and no concrete non‑discrete reflexive examples were known. The authors introduce the “P‑modification” of a topology: given a group G with its original topology τ, one defines a new topology τP by declaring every Gδ‑subset of (G,τ) to be open. This operation leaves the algebraic structure untouched while dramatically strengthening the topological properties.

The first major contribution is the construction of explicit non‑discrete reflexive P‑groups. The authors start with an infinite‑dimensional torus T^κ (κ a large cardinal) equipped with its usual compact product topology. Applying the P‑modification yields a group that is still a compact abelian group algebraically, but its topology is now a P‑topology: all Gδ‑sets are open, and the group is non‑discrete. The crucial observation is that the character group of this P‑modified torus coincides with the original character group of T^κ, which is again a torus of the same cardinality. Consequently the double dual (the Pontryagin bidual) is naturally isomorphic to the original P‑modified group, establishing reflexivity. This example settles the existence question in the affirmative.

The second major result concerns ω‑bounded groups, i.e., groups in which the closure of every countable subset is compact. The authors show that if an ω‑bounded abelian group G is equipped with its P‑modified topology, then G remains reflexive. The proof exploits the fact that ω‑boundedness guarantees that countable subsets have compact closures, and the P‑modification does not destroy this compactness. Moreover, the space of continuous characters on the P‑modified group is still sufficiently rich to separate points, allowing the Pontryagin duality machinery to go through unchanged. Thus non‑compact, non‑discrete ω‑bounded groups can be reflexive, contradicting earlier intuition that reflexivity required compactness or discreteness.

The central theorem of the paper is a generalization to arbitrary products of feathered (almost metrizable) abelian groups. A feathered group is one that contains a compact subgroup with a metrizable quotient; equivalently, it is almost metrizable. For any family {Ai}i∈I of feathered abelian groups, the authors consider the product ∏i Ai equipped with the product topology, then apply the P‑modification to this product. They prove that the resulting P‑product is reflexive. The argument proceeds in three steps:

1. Each factor Ai is shown to be reflexive after P‑modification, using the fact that feathered groups have a dense metrizable subspace and a well‑behaved character group.
2. The character group of the P‑product is identified with the direct product of the character groups of the factors, because the P‑modification does not interfere with the pointwise multiplication of characters.
3. Applying Pontryagin duality twice yields a natural isomorphism between the double dual and the original P‑product, establishing reflexivity.

A notable corollary is that every compact abelian group, when endowed with its P‑modified topology, is reflexive. Since compact groups are already reflexive in the classical sense, this result shows that reflexivity is robust under the drastic topological change induced by the P‑modification.

The paper resolves two previously posed problems. First, it answers positively the question of S. Hernández and P. Nickolas regarding the existence of non‑discrete reflexive P‑groups. Second, it solves the problem raised by Ardanza‑Trevijano, Chasco, Domínguez, and Tkachenko concerning the reflexivity of ω‑bounded groups.

Methodologically, the work blends classical Pontryagin duality, careful analysis of character groups, and the novel use of P‑modification to control the topology of Gδ‑sets. The authors also develop auxiliary lemmas concerning the preservation of density, weight, and compactness under P‑modification, which may be of independent interest.

In conclusion, the paper demonstrates that the P‑modification is a powerful tool for constructing new reflexive topological groups beyond the traditional discrete or compact realms. By providing explicit non‑discrete reflexive P‑groups, establishing reflexivity for ω‑bounded groups, and proving a broad product theorem for feathered groups, the authors significantly advance the understanding of reflexivity in topological group theory and open avenues for further exploration of P‑topologies in more exotic settings.