A survey of cardinality bounds on homogeneous topological spaces

TL;DR

This survey compiles and analyzes decades of results on bounds for the cardinality of homogeneous topological spaces, introduces new bounds, and discusses proof techniques and improvements.

Contribution

It provides a comprehensive overview of known bounds, introduces new cardinality bounds for homogeneous spaces, and discusses proof methods and recent generalizations.

Findings

Includes bounds like |X| ≤ 2^{πw(X)} and improvements for various classes.

Introduces new bounds involving πnχ(X) and qψ(X).

Provides a table of strongest known bounds and some proofs.

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 $|X|\leq 2^{\pi w(X)}$ if $X$ is a power homogeneous Hausdorff space, and its improvements $|X|\leq d(X)^{\pi\chi(X)}$ and $|X|\leq 2^{c(X)\pi\chi(X)}$ for spaces $X$ with the same properties. We also discuss de la Vega's Theorem, which states that $|X|\leq 2^{t(X)}$ 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) $|X|\leq d(X)^{\pi n\chi(X)}$ if $X$ is homogeneous and Hausdorff, and (2) $|X|\leq \pi\chi(X)^{c(X)q\psi(X)}$ if $X$ is a regular homogeneous space. The invariant $\pi n\chi(X)$, defined in this paper, has the property $\pi n\chi(X)\leq\pi\chi(X)$ and thus (1) improves the bound $d(X)^{\pi\chi(X)}$ for homogeneous Hausdorff spaces. The invariant $q\psi(X)$ has the properties $q\psi(X)\leq\pi\chi(X)$ and $q\psi(X)\leq\psi_c(X)$ if $X$ is Hausdorff, thus (2) improves the bound $2^{c(X)\pi\chi(X)}$ in the regular, homogeneous setting.