A survey of cardinality bounds on homogeneous topological spaces

Abstract

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| ≤ 2 πw(X) if X is a power homogeneous Hausdorff space [25], and its improvements |X| ≤ d(X) πχ(X) [42] and |X| ≤ 2 c(X)πχ(X)[18] for spaces X with the same properties. We also discuss de la Vega's Theorem, which states that |X| ≤ 2 t(X) if X is a homogeneous compactum [24], 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), defined in [32], has the properties qψ(X) ≤ πχ(X) and qψ(X) ≤ ψc(X) if X is Hausdorff, thus (2) improves the bound 2 c(X)πχ(X) in the regular, homogeneous setting.