Consistency and independence phenomena involving cellular-Lindelof spaces

Abstract

The cellular-Lindelöf property is a common generalization of the Lindelöf property and the countable chain condition that was introduced by Bella and Spadaro in 2018. We solve two questions of Alas, Gutierrez-Dominguez and Wilson by constructing consistent examples of a normal almost cellular-Lindelöf space which is neither cellular-Lindelöf nor weakly Lindelöf and a Tychonoff cellular-Lindelöf space of Lindelöf degree ${\omega }_1$ and uncountable weak Lindelöf degree for closed sets. We also construct a ZFC example of a space for which both the cellular-Lindelöf property and normality are undetermined in ZFC.