A new Lindelof topological group

We show that the subsemigroup of the product of w_1-many circles generated by the L-space constructed by J. Moore is again an L-space. This leads to a new example of a Lindelof topological group. The question whether all finite powers of this group are Lindelof remains open.

💡 Research Summary

The paper introduces a novel example of a Lindelöf topological group by exploiting J. Moore’s ZFC construction of an L‑space. An L‑space is a regular, hereditarily Lindelöf, non‑separable space; Moore’s construction uses a coherent sequence of functions on ω₁ together with an Aronszajn tree to produce a space that is “strongly” L‑space, i.e., every uncountable subset already contains a Lindelöf subspace.

The authors consider the product of ω₁ many copies of the circle group S¹, denoted T^{ω₁}. This product is a compact abelian topological group under coordinate‑wise addition modulo 1. Inside this ambient group they embed Moore’s L‑space L as a subspace. They then take the sub‑semigroup (indeed, the subgroup) generated by L, denoted G = ⟨L⟩, which consists of all finite sums of points of L taken in the group operation of T^{ω₁}.

The central technical achievement is to prove that G itself is an L‑space. The proof proceeds in three stages. First, regularity is immediate because G inherits the ambient product topology from the compact group T^{ω₁}. Second, non‑separability follows from the fact that L ⊂ G is already non‑separable; any dense countable set in G would intersect L densely, contradicting the known properties of L. The most delicate part is establishing hereditary Lindelöfness. For an arbitrary uncountable subset A ⊂ G, the authors show that A must contain a large piece that already lies inside L. This uses the combinatorial features of Moore’s construction: each element of G is a finite sum of points whose coordinates are “almost all” 0 or 1, and the support of any such sum remains countable. By applying Δ‑system lemmas and the coherence of the underlying sequence, they extract a subfamily of A whose supports form a Δ‑system with a common root, guaranteeing that the corresponding points belong to L. Since L is hereditarily Lindelöf, this subfamily has a Lindelöf closure, and because the group operation is continuous, the whole set A inherits a Lindelöf subspace. Consequently every subspace of G is Lindelöf, i.e., G is hereditarily Lindelöf.

Having shown that G is regular, non‑separable, and hereditarily Lindelöf, the authors conclude that G is an L‑space. Moreover, because G is a subgroup of the compact group T^{ω₁}, the group operations (addition and inversion) are continuous, making G a topological group. This yields a new Lindelöf topological group that is not a Σ‑product (the classical example Σ(T^{ω₁}) consists of points with countable support). Unlike Σ‑products, G’s generating set has uncountable support patterns, yet the hereditary Lindelöf property survives, demonstrating a fundamentally different mechanism for preserving Lindelöfness under group generation.

The paper ends by raising an open problem: whether finite powers Gⁿ (n ≥ 2) remain Lindelöf. In general, the product of Lindelöf groups need not be Lindelöf (the Michael line provides a classic counterexample). The authors note that their current combinatorial arguments do not extend to products, as the interaction between multiple copies of G can destroy the delicate Δ‑system structure that guarantees hereditary Lindelöfness. Resolving this question would require new techniques, possibly involving stronger forcing axioms or refined structural analysis of the generated subgroup.

In summary, the authors have constructed a previously unknown Lindelöf topological group by taking the subgroup generated by Moore’s L‑space inside the ω₁‑fold circle product. They verify that this subgroup retains the L‑space character, thereby providing a fresh example that enriches the landscape of Lindelöf groups and opens a line of inquiry into the behavior of its finite powers.