We describe several ordinal indices that are capable of detecting, according to various metric notions of faithfulness, the embeddability between pairs of Polish spaces. These embeddability ranks are of theoretical interest but seem difficult to estimate in practice. Embeddability ranks, which are easier to estimate in practice, are embeddability ranks generated by Schauder bases. These embeddability ranks are inspired by the nonlinear indices à la Bourgain. In particular, we resolve a problem raised by F. Baudier, G. Lancien, P. Motakis, and Th. Schlumprecht in Coarse and Lipschitz universality, Fund. Math. 254 (2021), no. 2, 181–214, regarding the necessity of additional set-theoretic axioms regarding the main coarse universality result there.

中文翻译：

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抽象可嵌入性排名

我们描述了几个序数索引，它们能够根据各种忠实度度量概念来检测波兰空间对之间的可嵌入性。这些可嵌入性等级具有理论上的意义，但在实践中似乎很难估计。嵌入性排名在实践中更容易估计，是由 Schauder 基生成的嵌入性排名。这些可嵌入性排名的灵感来自于布尔干的非线性指数。特别是，我们解决了 F. Baudier、G. Lancien、P. Motakis 和 Th 提出的问题。粗略的 Schlumprecht 和 Lipschitz 普遍性，基金。数学。 254（2021），没有。 2, 181-214，关于关于主要粗略普遍性结果的附加集合论公理的必要性。