LAUSR

Abstract In this paper, we make some observations on chain conditions and dually properties. In particular, we show that: (1) A subspace X ⊂ ω 1 ω is dually CCC then e ( X ) ≤ ω and a normal subspace X ⊂ ω 1 ω is DCCC if and only if e ( X ) ≤ ω ; (2) There is a Tychonoff pseudocompact subspace X ⊂ ( ω 1 + 1 ) 2 which is not dually CCC; (3) In the class of o-semimetrizable spaces, dually separable is self-dual with respect to neighbourhood assignments. As an application, we obtain an example of a CCC normal Moore space which is not dually separable under MA+¬CH; (4) There exists an example of a large normal CCC semi-stratifiable space, which answers a question of Xuan and Song (2018) [21, Question 4.11] ; (5) Every dually separable and monotonically monolithic space is Lindelof, which gives a partial answer to a question of Alas, Junqueira, van Mill, Tkachuk and Wilson (2011) [2, Question 2.1] ; (6) A dually separable Hausdorff space with a strong rank 1-diagonal has cardinality at most 2 c . The conclusion is also true for regular spaces if we replace “strong rank 1-diagonal” with “ G δ -diagonal”; (7) A dually separable ω-monolithic Hausdorff space with a G δ -diagonal has cardinality at most c .