Cardinality of cellular-Lindelöf first-countable spaces

Determine whether every cellular-Lindelöf first-countable topological space, and in particular every regular cellular-Lindelöf first-countable space, has cardinality at most the continuum.

Background

The paper studies variants of the Lindelöf property involving discrete and strongly discrete sets. Cellular-Lindelöf spaces are those in which every pairwise disjoint family of non-empty open sets meets a Lindelöf subspace. Arhangel’skii-type cardinality bounds are central to this line of inquiry.

Prior results show strong cardinality bounds for related classes (e.g., strongly discretely Lindelöf, almost discretely Lindelöf, and cellular-compact spaces under additional hypotheses), but a general bound for cellular-Lindelöf first-countable spaces remains unresolved. This question serves as a fundamental open problem motivating several partial results in the paper.