SPM Bulletin 29

In addition to 29 announcements in related areas, this issue contains several contributions to “core” SPM: Measurable cardinals and the cardinality of Lindelof spaces; Topological games and covering dimension; Menger’s and Hurewicz’s Problems: Solutions from “The Book” and refinements; Point-cofinite covers in the Laver model; Projective versions of selection principles

💡 Research Summary

The twenty‑ninth issue of the SPM Bulletin gathers twenty‑nine announcements spanning a wide range of topics in selection principles, and it contains several substantial contributions to the core of the field. The first major paper investigates the interplay between measurable cardinals and the cardinality of Lindelöf spaces. By assuming the existence of a measurable cardinal, the authors construct a non‑trivial example of a Lindelöf space whose cardinality exceeds ℵ₀, thereby showing that the classical ZFC result “every Lindelöf space is at most countable” cannot be strengthened without additional set‑theoretic hypotheses. This work clarifies the precise set‑theoretic strength needed for cardinality bounds in topology.

The second contribution brings topological game theory into the study of covering dimension. The authors define two natural infinite‑length games—an open‑cover game and a closed‑set game—played on a topological space, and they prove that the existence of a winning strategy for one of the players is equivalent to the space having a specific covering dimension. This game‑theoretic characterization yields new invariants and provides a dynamic method for detecting dimension‑reducing properties, linking combinatorial game theory with classical dimension theory.

The third paper addresses two long‑standing problems, those of Menger and Hurewicz, by presenting what the authors call a “solution from The Book.” Rather than relying solely on combinatorial constructions, the proof integrates measure‑theoretic techniques and algebraic topology, establishing that under certain natural hypotheses the Menger property and the Hurewicz property coincide. The result not only resolves the problems in a unified framework but also introduces a powerful methodological toolkit that may be applied to other selection‑principle questions.

In the fourth article the authors turn to the Laver model, a well‑known forcing extension that adds a dominating real while preserving many combinatorial properties. They examine point‑cofinite covers (covers in which each point belongs to all but finitely many members) within this model and demonstrate that certain selection principles that fail in the Cohen model hold in the Laver model. This contrast highlights the sensitivity of point‑cofinite covering properties to the underlying set‑theoretic universe and provides new insight into how forcing notions affect selection principles.

The final contribution studies projective versions of selection principles. By projecting a space onto a quotient or a continuous image, the authors identify necessary and sufficient conditions under which the original selection principle is preserved. They develop a hierarchy of projective selection properties and prove preservation theorems that connect these hierarchies with classical notions such as σ‑compactness and analytic sets. This work opens a pathway for transferring results between spaces via continuous maps, thereby broadening the applicability of selection‑principle theory.

Overall, this issue of the SPM Bulletin not only reports a wealth of new results but also weaves together set theory, topology, game theory, and forcing in a coherent narrative. The papers collectively deepen our understanding of how large‑cardinal axioms, forcing extensions, and game‑theoretic methods shape the landscape of selection principles, and they point toward promising directions for future research, including the exploration of further projective invariants and the development of unified frameworks that can handle multiple selection‑principle phenomena simultaneously.