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

TL;DR

This paper establishes the equivalence of various topological properties like compactness and convergence through filters, and characterizes product spaces' compactness and related properties using set-theoretic invariants.

Contribution

It provides new characterizations of product spaces' compactness and convergence properties using filter-based notions and set-theoretic invariants such as the splitting and distributivity numbers.

Findings

Sequential compactness of products depends on subproducts with fewer factors.

Product is Lindelöf if all subproducts with countably many factors are Lindelöf.

Parallel results for other compactness and covering properties.

Abstract

We show that, under suitably general formulations, covering properties, accumulation properties and filter convergence are all equivalent notions. This general correspondence is exemplified in the study of products. Let $X$ be a product of topological spaces. We prove that $X$ 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 ${\rm cf} \mathfrak s \geq \mathfrak h$. Recall that $\mathfrak s$ denotes the splitting number and $\mathfrak h$ the distributivity number. The product $X$ is Lindel\"of if and only if all subproducts by $\leq \omega_1 $ factors are Lindel\"of. Parallel results are obtained for final $ \omega_n$-compactness, $[ \lambda, \mu ]$-compactness, the Menger and Rothberger properties.