Consistency and independence phenomena involving cellular-Lindelof spaces

TL;DR

This paper explores the properties of cellular-Lindel"of spaces, providing new examples and resolving open questions about their consistency, normality, and Lindel"of degrees within set-theoretic frameworks.

Contribution

It constructs novel examples of cellular-Lindel"of spaces with specific properties and addresses open questions about their consistency and normality in ZFC.

Findings

Constructed a normal almost cellular-Lindel"of space not cellular-Lindel"of or weakly Lindel"of.

Created a Tychonoff cellular-Lindel"of space with high Lindel"of degree and uncountable weak Lindel"of degree.

Provided a ZFC example where cellular-Lindel"of property and normality are independent.

Abstract

The cellular-Lindel\"of property is a common generalization of the Lindel\"of 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-Lindel\"of space which is neither cellular-Lindel\"of nor weakly Lindel\"of and a Tychonoff cellular-Lindel\"of space of Lindel\"of degree $\omega_1$ and uncountable weak Lindel\"of degree for closed sets. We also construct a ZFC example of a space for which both the cellular-Lindel\"of property and normality are undetermined in ZFC.