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Keywords:
filter convergence; ultrafilter; product; subproduct; sequential compactness; sequencewise $\mathcal P$-compactness; Lindelöf property; final $\lambda$-compactness; $[ \mu, \lambda ]$-compactness; Menger property; Rothberger property
filter convergence; ultrafilter; product; subproduct; sequential compactness; sequencewise $\mathcal P$-compactness; Lindelöf property; final $\lambda$-compactness; $[ \mu, \lambda ]$-compactness; Menger property; Rothberger property
Summary:
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. We prove that a product is Lindelöf if and only if all subproducts by $\leq \omega_1 $ factors are Lindelöf. Parallel results are obtained for final $ \omega_n$-compactness, $[ \lambda, \mu ]$-compactness, the Menger and the Rothberger properties.
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