Notes on star Lindelöf space

In this paper, we prove that the cardinality of a star Lindelöf space X   does not exceed cc if X satisfies one of the following conditions: (1) X has a rank 3-diagonal; (2) X is normal and has a rank 2-diagonal; (3) X   is first countable, normal and has a GδGδ-diagonal. Moreover, we also obtain several results concerning the general question “When must a star Lindelöf space be star countable?”.