Cardinal invariants of cellular Lindelof spaces

A space X is said to be "cellular-Lindelöf" if for every cellular family  there is a Lindelöf subspace L of X which meets every element of . 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 2. We also prove that every normal cellular-Lindelöf first-countable space has cardinality at most continuum under 2<= and that every normal cellular Lindelöf space with a Gδ-diagonal of rank 2 has cardinality at most continuum.