2023/09/12 by Ziqin Feng, Feng, Ziqin, Paul Gartside +1
Computer Science · #Topological and Geometric Data Analysis #Computability, Logic, AI Algorithms
paper · pdf · doi:10.48550/arxiv.2309.06523
Every directed set is Tukey equivalent to (a) the family of all compact subsets, ordered by inclusion, of a (locally compact) space, to (b) a neighborhood filter, ordered by reverse inclusion, of a point (of a compact space, and of a topological group), and to (c) the universal uniformity, ordered by reverse inclusion, of a space. Two directed sets are Tukey equivalent if they are cofinally equivalent in the sense that they can both be order embedded cofinally in a third directed set. In contrast, any totally bounded uniformity is Tukey equivalent to [κ]<ω, the collection of all finite subsets of κ, where κ is the cofinality of the uniformity. All other Tukey types are `rejected' by totally bounded uniformities. Equivalently, a compact space X has weight (minimal size of a base) equal to κ if and only if the neighborhood filter of the diagonal is Tukey equivalent to [κ]<ω. A number of questions from the literature are answered with the aid of the above results.