Productively Lindel\"of spaces of countable tightness

TL;DR

This paper proves that under the Continuum Hypothesis, productively Lindel"of spaces with countable tightness are powerfully Lindel"of, extending previous results and providing new proofs and insights into the structure of such spaces.

Contribution

It establishes that under CH, productively Lindel"of spaces of countable tightness are powerfully Lindel"of, and shows separation axioms are not essential for counterexamples to Michael's question.

Findings

Under CH, productively Lindel"of spaces of countable tightness are powerfully Lindel"of.

New proofs of results by Arkhangel'skii and Buzyakova.

Separation axioms are not relevant to Michael's question.

Abstract

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.