SPM Bulletin 24

1. A Wikipedia entry on topological games 2. On a fragment of the universal Baire property for sigma^1_2 sets 3. The coarse classification of homogeneous ultra-metric spaces 4. Ramsey-like embeddings 5. Proper and piecewise proper families of reals 6. Measures and their random reals 7. Obtainable Sizes of Topologies on Finite Sets 8. Spaces of R-places of rational function fields 9. All properties in the Scheepers Diagram are linearly-sigma-additive.

💡 Research Summary

**
The March 2008 issue of the SPM Bulletin (Issue 24) presents nine research announcements spanning set theory, topology, measure theory, and model theory, followed by a list of twenty‑two open problems from earlier issues.

1. A Wikipedia entry on topological games – This announcement points to a newly created Wikipedia article that surveys the theory of topological games, such as the classic open‑cover (O, Ω) and selection games. By providing definitions, key theorems, and references, the entry serves as a bridge between specialists and a broader audience, encouraging wider dissemination of the subject.

2. On a fragment of the universal Baire property for Σ¹₂ sets – The author proves that under modest set‑theoretic assumptions (far weaker than full projective determinacy) a substantial fragment of the universal Baire property holds for Σ¹₂ sets. The result links descriptive set theory with forcing and large‑cardinal techniques, showing that many Σ¹₂ sets are “large’’ both in the Baire category sense and in measure‑theoretic terms.

3. The coarse classification of homogeneous ultra‑metric spaces – Banakh and Zarichnyy introduce the invariant “sharp entropy’’ Ent♯(X) for a homogeneous ultra‑metric space X. They prove the exact classification theorem: two such spaces are coarsely equivalent iff their sharp entropies coincide. Consequently every homogeneous ultra‑metric space is coarsely equivalent to the anti‑Cantor set 2 < ω. The proof relies on a novel “tower’’ construction that may have independent interest for coarse geometry.

4. Ramsey‑like embeddings – Gitman defines a hierarchy of large‑cardinal axioms extending the classical notion of Ramsey embeddings. By formulating new elementary embedding schemes for transitive sets of size κ, she shows that these axioms sit strictly between weak compactness and measurability. This work refines the landscape of large cardinals and provides fresh combinatorial characterizations of Ramsey‑type strength.

5. Proper and piecewise proper families of reals – Also by Gitman, this note introduces two new notions of families of reals. A family X is proper if it is arithmetically closed and the quotient Boolean algebra X/fin is a proper poset; piecewise proper families are unions of ω₁‑length chains of proper families. The paper investigates the existence of such families of various cardinalities, motivated by the open problem of whether every Scott set can be realized as the standard system of a model of PA. The results give new tools for constructing models of arithmetic with prescribed standard systems.

6. Measures and their random reals – Reimann and Slaman study randomness relative to arbitrary probability measures on Cantor space. They prove two complementary facts: (i) every non‑recursive real is random for some (possibly atomic) measure, and (ii) every non‑hyperarithmetical real is random for some continuous (non‑atomic) measure. Conversely, they exhibit hyperarithmetical reals that fail to be random for any continuous measure, locating the boundary between algorithmic randomness and descriptive set‑theoretic complexity.

7. Obtainable sizes of topologies on finite sets – Ragnarsson and Tenn­er analyze the minimal number of points required for a finite topological space to have exactly k open sets (equivalently, a poset with k order ideals). Using efficient constructive algorithms they obtain a logarithmic upper bound on this minimal size and show that for each n there exists a topology on n points with k open sets for a whole interval of k that grows exponentially in n. They also adapt the construction to control the size of the smallest neighbourhood of each point, thereby describing a wide range of attainable topology sizes.

8. Spaces of R‑places of rational function fields – Machura and Osiak answer a question about when different orderings of the rational function field R(X) (R a real‑closed field) induce the same real place. Their analysis shows that the space of R‑places of the field R(Y) (where R is any real closure of R(X)) is not metrizable. Consequently the space M(R(X,Y)) fails to be metrizable, illustrating the topological complexity of spaces of real places.

9. All properties in the Scheepers Diagram are linearly‑σ‑additive – Building on Jordan’s recent theorem that S₁(Ω, Γ) is linearly‑σ‑additive, the authors (Tsaban, Zdomskyy, and others) provide an elementary combinatorial proof that this preservation property holds for every selection principle appearing in the Scheepers Diagram, including those that are not σ‑additive in the usual sense (e.g., S₁(Γ, Ω) and S_fin(Γ, Ω)). This settles several open questions (e.g., Problem 4.9 in