  Michael asked whether every productively Lindel\"of space is powerfully
Lindel\"of. Building of work of Alster and De la Vega, assuming the Continuum
Hypothesis, we show that every productively Lindel\"of space of countable
tightness is powerfully Lindel\"of. This strengthens a result of Tall and
Tsaban. The same methods also yield new proofs of results of Arkhangel'skii and
Buzyakova. Furthermore, assuming the Continuum Hypothesis, we show that a
productively Lindel\"of space $X$ is powerfully Lindel\"of if every open cover
of $X^\omega$ admits a point-continuum refinement consisting of basic open
sets. This strengthens a result of Burton and Tall. Finally, we show that
separation axioms are not relevant to Michael's question: if there exists a
counterexample (possibly not even $\mathsf{T}_0$), then there exists a regular
(actually, zero-dimensional) counterexample.

General Topology;Logic