2020/07/28 by Nathan Carlson, Carlson, Nathan
Decision Sciences · Mathematics · #Advanced Banach Space Theory #Advanced Topology and Set Theory #FOS: Mathematics #Fuzzy and Soft Set Theory #General Topology (math.GN)
paper · pdf · doi:10.48550/arxiv.2007.14326
openalex publication_date 2020/07/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this survey we catalogue the many results of the past several decades concerning bounds on the cardinality of a topological space with homogeneous or homogeneous-like properties. These results include van Douwen's Theorem, which states |X|≤ 2πw(X) if X is a power homogeneous Hausdorff space, and its improvements |X|≤ d(X)πχ(X) and |X|≤ 2c(X)πχ(X) for spaces X with the same properties. We also discuss de la Vega's Theorem, which states that |X|≤ 2t(X) if X is a homogeneous compactum, as well as its recent improvements and generalizations to other settings. This reference document also includes a table of strongest known cardinality bounds on spaces with homogeneous-like properties. The author has chosen to give some proofs if they exhibit typical or fundamental proof techniques. Finally, a few new results are given, notably (1) |X|≤ d(X)πnχ(X) if X is homogeneous and Hausdorff, and (2) |X|≤ πχ(X)c(X)qψ(X) if X is a regular homogeneous space. The invariant πnχ(X), defined in this paper, has the property πnχ(X)≤πχ(X) and thus (1) improves the bound d(X)πχ(X) for homogeneous Hausdorff spaces. The invariant qψ(X) has the properties qψ(X)≤πχ(X) and qψ(X)≤ψc(X) if X is Hausdorff, thus (2) improves the bound 2c(X)πχ(X) in the regular, homogeneous setting.