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|≤ 2π w(X) if X is a power homogeneous Hausdorff space, and its improvements |X|≤ d(X)π(X) and |X|≤ 2c(X)π(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)π n(X) if X is homogeneous and Hausdorff, and (2) |X|≤ π(X)c(X)q(X) if X is a regular homogeneous space. The invariant π n(X), defined in this paper, has the property π n(X)≤π(X) and thus (1) improves the bound d(X)π(X) for homogeneous Hausdorff spaces. The invariant q(X) has the properties q(X)≤π(X) and q(X)≤c(X) if X is Hausdorff, thus (2) improves the bound 2c(X)π(X) in the regular, homogeneous setting.