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

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 $\le \mathfrak{s}$ 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 $\mathrm{cf}\mathfrak{s}\ge \mathfrak{h}$. Recall that $\mathfrak{s}$ denotes the splitting number and $\mathfrak{h}$ the distributivity number. The product $X$ is Lindelöf if and only if all subproducts by $\le {\omega }_1$ factors are Lindelöf. Parallel results are obtained for final ${\omega }_n$-compactness, $[\lambda ,\mu ]$-compactness, the Menger and Rothberger properties.

v3, entirely rewritten with many additions; v4, fixed some details; v5, a few improvements