Cardinal invariants of cellular Lindelof spaces

A space X is said to be cellular-Lindelöf if for every cellular family U there is a Lindelöf subspace L of X which meets every element of U. Cellular-Lindelöf spaces generalize both Lindelöf spaces and spaces with the countable chain condition. Solving questions of Xuan and Song, we prove that every cellular-Lindelöf monotonically normal space is Lindelöf and that every cellular-Lindelöf space with a regular Gδ -diagonal has cardinality at most 2c. We also prove that every normal cellular-Lindelöf first-countable space has cardinality at most continuum under 2