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A survey of cardinality bounds on homogeneous topological spaces

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 $|X|\leq 2^{πw(X)}$ if $X$ is a power homogeneous Hausdorff space, and its improvements $|X|\leq d(X)^{πχ(X)}$ and $|X|\leq 2^{c(X)πχ(X)}$ 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) $|X|\leq d(X)^{πnχ(X)}$ if $X$ is homogeneous and Hausdorff, and (2) $|X|\leq πχ(X)^{c(X)qψ(X)}$ if $X$ is a regular homogeneous space. The invariant $πnχ(X)$, defined in this paper, has the property $πnχ(X)\leqπχ(X)$ and thus (1) improves the bound $d(X)^{πχ(X)}$ for homogeneous Hausdorff spaces. The invariant $qψ(X)$ has the properties $qψ(X)\leqπχ(X)$ and $qψ(X)\leqψ_c(X)$ if $X$ is Hausdorff, thus (2) improves the bound $2^{c(X)πχ(X)}$ in the regular, homogeneous setting.