A characterization of the Menger property by means of ultrafilter convergence

We characterize various Menger/Rothberger related properties by means of ultrafilter convergence, and discuss their behavior with respect to products.Comment: v.2; 13 pages, some improvements, some corrections, added introduction and divided into section

Survey on the Tukey theory of ultrafilters

This article surveys results regarding the Tukey theory of ultrafilters on countable base sets. The driving forces for this investigation are Isbell's Problem and the question of how closely related the Rudin-Keisler and Tukey reducibilities are. We review work on the possible structures of cofinal types and conditions which guarantee that an ultrafilter is below the Tukey maximum. The known canonical forms for cofinal maps on ultrafilters are reviewed, as well as their applications to finding which structures embed into the Tukey types of ultrafilters. With the addition of some Ramsey theory, fine analyses of the structures at the bottom of the Tukey hierarchy are made.Comment: 25 page

Products of sequentially compact spaces and compactness with respect to a set of filters

Let $X$ be a product of topological spaces. $X$ is sequentially compact if and only if all subproducts by $\leq \mathfrak s$ factors are sequentially compact. If $\mathfrak s = \mathfrak h$, then $X$ is sequentially compact if and only if all factors are sequentially compact and all but at most $<\mathfrak s$ factors are ultraconnected. We give a topological proof of the inequality $cf \mathfrak s \geq \mathfrak h$. Recall that $\mathfrak s$ denotes the splitting number, and $\mathfrak h$ the distributivity number. Parallel results are obtained for final $\omega_n$-compactness and for other properties, as well as in the general context of a formerly introduced notion of compactness with respect to a set of filters. Some corresponding invariants are introduced.Comment: v3, entirely rewritten with many additions v4, fixed some detail