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- W4386793398 abstract "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 $[kappa]^{<omega}$, the collection of all finite subsets of $kappa$, where $kappa$ 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 $kappa$ if and only if the neighborhood filter of the diagonal is Tukey equivalent to $[kappa]^{<omega}$. A number of questions from the literature are answered with the aid of the above results." @default.
- W4386793398 created "2023-09-16" @default.
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- W4386793398 date "2023-09-12" @default.
- W4386793398 modified "2023-09-30" @default.
- W4386793398 title "Directed Sets of Topology -- Tukey Representation and Rejection" @default.
- W4386793398 doi "https://doi.org/10.48550/arxiv.2309.06523" @default.
- W4386793398 hasPublicationYear "2023" @default.
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