Weak covering properties and selection principles

No convenient internal characterization of spaces that are productively Lindelöf is known. Perhaps the best general result known is Alsterʼs internal characterization, under the Continuum Hypothesis, of productively Lindelöf spaces which have a basis of cardinality at most ℵ1ℵ1. It turns out that topological spaces having Alsterʼs property are also productively weakly Lindelöf. The weakly Lindelöf spaces form a much larger class of spaces than the Lindelöf spaces. In many instances spaces having Alsterʼs property satisfy a seemingly stronger version of Alsterʼs property and consequently are productively X, where X is a covering property stronger than the Lindelöf property. This paper examines the question: When is it the case that a space that is productively X is also productively Y, where X and Y are covering properties related to the Lindelöf property.