A study of chain conditions and dually properties

Abstract

In this paper, we make some observations on chain conditions and dually properties. In particular, we show that:(1) A subspace X⊂ω1ω is dually CCC then e(X)≤ω and a normal subspace X⊂ω1ω is DCCC if and only if e(X)≤ω;(2) There is a Tychonoff pseudocompact subspace X⊂(ω1+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 Lindelöf, 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 2c. The conclusion is also true for regular spaces if we replace “strong rank 1-diagonal” with “Gδ-diagonal”;(7) A dually separable ω-monolithic Hausdorff space with a Gδ-diagonal has cardinality at most c.