Tukey Order, Calibres and the Rationals

TL;DR

This paper investigates the Tukey order structure of the set of compact subsets of various topological spaces, especially focusing on the space of rationals, and derives invariants and bounds for these structures.

Contribution

It computes Tukey bounds and invariants for the compact sets of certain spaces, notably the rationals, and analyzes their structure under the Tukey order.

Findings

Calculated Tukey bounds for $\\mathcal{K}(M)$ in specific spaces.

Determined invariants of the compact set structures.

Analyzed the Tukey order position of $\\mathcal{K}(M)$ for various classes of spaces.

Abstract

One partially ordered set, $Q$, is a Tukey quotient of another, $P$, denoted $P \geq_T Q$, if there is a map $\phi : P \to Q$ carrying cofinal sets of $P$ to cofinal sets of $Q$. Let $X$ be a space and denote by $\mathcal{K}(X)$ the set of compact subsets of $X$, ordered by inclusion. For certain separable metrizable spaces $M$, Tukey upper and lower bounds of $\mathcal{K}(M)$ are calculated. Results on invariants of $\mathcal{K}(M)$'s are deduced. The structure of all $\mathcal{K}(M)$'s under $\le_T$ is investigated. Particular emphasis is placed on the position of $\mathcal{K}(M)$ when $M$ is: completely metrizable, the rationals $\mathbb{Q}$, co-analytic or analytic.