The Cofinality of Generating Familes (joint with Paul Gartside)

Cofinal sets, which study what happens “eventually” in a given order, are ubiquitous in mathematics. In Topology, they play a particularly noticeable role. Consider, for example, a subspace $M$ of the reals. The topology can be captured by convergent sequences, and hence we can ask the question: how many sequences are needed to characterize the topology on $M$? As another example, consider questions (essentially about cofinalities) such as: how many compact sets cover $M$ (the “compact covering number” of $M$)? or how many compact subsets generate the topology on $M$ (the $k$-ness number of $M$)?

In this talk, we will discuss recent work of Paul Gartside and the speaker on these questions. We will show how these questions about how many sequences (or compact sets) generate the topology of $M$ can be viewed through the lens of the Tukey order on relations and how, as such, we can apply set-theoretic techniques. We show how certain cardinal invariants are used to answer these topological questions. Finally, we discuss how (in light of recent work of James Cummings and the speaker on extender-based forcing and mod-finite scales) we can, among spaces with a fixed value of the compact covering number, the $k$-ness number can be quite varied.