Directed sets of topology: Tukey representation and rejection

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.