Directed Sets of Topology -- Tukey Representation and Rejection

TL;DR

This paper classifies directed sets up to Tukey equivalence, linking them to various topological and uniform structures, and explores the implications for compact spaces and their weights.

Contribution

It provides a comprehensive classification of directed sets via Tukey equivalence, connecting them to topological and uniform structures, and clarifies the concept of rejection of certain Tukey types.

Findings

Directed sets are Tukey equivalent to families of compact subsets, neighborhood filters, or universal uniformities.

Totally bounded uniformities are Tukey equivalent to finite subsets of a cardinal, and others are rejected.

A compact space's weight equals the minimal size of a base if its diagonal neighborhood filter is Tukey equivalent to finite subsets of a cardinal.

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.