论文标题
对均质拓扑空间的基数界限的调查
A survey of cardinality bounds on homogeneous topological spaces
论文作者
论文摘要
在这项调查中，我们将过去几十年的许多结果分类，这些结果涉及具有均匀或同质性特性的拓扑空间的基数。这些结果包括van douwen的定理，该定理指出$ | x | \ leq 2^{πw（x）} $如果$ x $是均质的均值Hausdorff空间，其改进$ | x | x | \ leq d（x）^{π取（πt（x）} $ | x | x | x | x | \ leq 2^$ q osq 2^c（x） 特性。我们还讨论了de la vega的定理，该定理指出，如果$ | x | \ leq 2^{t（x）} $如果$ x $是一种同质的契约，以及其最近对其他设置的改进和概括。该参考文档还包括具有均匀特性的空间上最强的已知基数边界表。作者选择了一些证据，如果它们表现出典型或基本的证明技术。最后，值得注意的是（1）$ | x | \ leq d（x）^{πnχ（x）} $，如果$ x $是均匀的，而Hausdorff和（2）$ | x | x | \ leq feqπχ（x）^{c（x）q（x）q 6 $ x $是$ x $是常规的家，则是常规的空间。本文定义的不变$πnχ（x）$具有属性$πnχ（x）\leqπχ（x）$，因此（1）改善了均质的hausdorff Space的绑定$ d（x）^{πχ（x）} $。不变的$qψ（x）$具有属性$qψ（x）\leqπχ（x）$和$qψ（x）\leqψ_c（x）$如果$ x $是hausdorff，则（2）在常规，同型设置中改善绑定的$ 2^{c（x）π取（x）π取（x）π取（x）π取。
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.
