  No convenient internal characterization of spaces that are productively
Lindelof is known. Perhaps the best general result known is Alster's internal
characterization, under the Continuum Hypothesis, of productively Lindelof
spaces which have a basis of cardinality at most $\aleph_1$. It turns out that
topological spaces having Alster's property are also productively weakly
Lindelof. The weakly Lindelof spaces form a much larger class of spaces than
the Lindelof spaces. In many instances spaces having Alster's property satisfy
a seemingly stronger version of Alster's property and consequently are
productively X, where X is a covering property stronger than the Lindelof
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 Lindelof property.
