A study of chain conditions and dually properties

摘要

In this paper, we make some observations on chain conditions and dually properties. In particular, we show that:(1) A subspace X subset of omega(omega)(1) is dually CCC then e(X) = omega and a normal subspace X subset of omega(omega)(1) is DCCC if and only if e(X) = omega;(2) There is a Tychonoff pseudocompact subspace X subset of (omega(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+(sic)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(delta)-diagonal";(7) A dually separable omega-monolithic Hausdorff space with a G(delta)-diagonal has cardinality at most c. (C) 2018 Elsevier B.V. All rights reserved.