Dually properties and cardinal inequalities

In this paper, we prove that DCCC is self-dual with respect to neighborhood assignments. Some related results on dually CCC (or dually DCCC) spaces are also obtained. Moreover, we prove that the cardinality of a dually CCC space X is at most 2ω if X satisfies one of the following conditions: (1) X has a rank 2-diagonal; (2) X is first countable and has a Gδ-diagonal; (3) X is first countable and perfect. Finally, we prove that the extent of a first countable dually CCC space is at most 2ω.