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Abstract Given a topological property (or a class) P , the class P ′ consists of spaces X such that for any neighbourhood assignment ϕ on X , there exists a subspace Y ⊂ X with property P for which ϕ (Y) = ⋃ { ϕ (y) : y ∈ Y } is dense in X. The class P ′ are called the weak dual of P or weakly dually P (with respect to neighbourhood assignments). We establish that DCCC is weakly self-dual in the class of weakly regular spaces. If P ∈ { weakly Lindelöf , CCC , separable } , then P is weakly self-dual in the class of Baire developable spaces. By using Erdös–Radó's theorem, we also prove that: (1) If X is a Baire, weakly dually CCC Hausdorff space with a rank 2-diagonal, then X has cardinality at most 2 ω ; (2) If X is a Baire, weakly dually DCCC Hausdorff space with a rank 3-diagonal, then X has cardinality at most 2 ω ; (3) If X is a weakly dually DCCC Hausdorff space with a rank 4-diagonal, then X has cardinality at most 2 ω. [ABSTRACT FROM AUTHOR]

Abstract Whenever P is a topological property, we say that a topological space X is star P if whenever U is an open cover of X , there is a subspace Y ⊂ X with property P such that St (Y , U) = X. We study the relationships of star P properties for P ∈ { DCCC , weakly Lindelöf , CCC } with other chain conditions. By using Erdös–Radó's theorem, we also establish several cardinal inequalities for star CCC spaces. [ABSTRACT FROM AUTHOR]

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 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 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. [ABSTRACT FROM AUTHOR]

Abstract The class of cellular-Lindelöf spaces was introduced and studied by A. Bella and S. Spadaro (2017) [4]. We say that a topological space X is cellular-Lindelöf if for every family U of pairwise disjoint non-empty open sets of X there is a Lindelöf subspace L ⊂ X such that U ∩ L ≠ ∅ , for every U ∈ U. In this paper, we first study topological properties of cellular-Lindelöf spaces, and the relations between cellular-Lindelöf spaces and related spaces. In particular, we obtain a Tychonoff example of a weakly Lindelöf space which is not cellular-Lindelöf, which gives a positive answer to a question of A. Bella and S. Spadaro ([4, Question 2]). We also prove that every monotonically normal W -space is cellular-Lindelöf if and only if it is Lindelöf. Finally, by using Erdös–Radó's theorem, we establish some cardinal inequalities for cellular-Lindelöf spaces. Some new questions are also posed. [ABSTRACT FROM AUTHOR]

We prove that a Hausdorff metaLindelöf weakly star countable space is feebly Lindelöf and a Hausdorff metacompact weakly star finite space is almost compact which partially answers a question of Alas and Wilson (2017) [2, Question 3.14] . We also obtain a normal example of a weakly star countable space which is neither almost star countable nor star Lindelöf without any set-theoretic assumptions, which answers a question implicitly asked by Song (2015) [13, Remark 2.8] and a question asked by Alas, Junqueira and Wilson (2011) [3, Question 4] . Under MA+¬CH, there even exists a normal weakly star countable Moore space which is not almost star countable. An example of a Tychonoff star compact and weakly star finite space which is not star countable is also given. Finally, we prove that every weakly star countable Hausdorff space with a rank 4-diagonal has cardinality at most 2 ω . [ABSTRACT FROM AUTHOR]

We say that a space X is dually CCC (respectively, weakly Lindelöf, separable) if for any neighbourhood assignment ϕ on X , there is a CCC (respectively, weakly Lindelöf, separable) subspace Y ⊂ X such that ϕ (Y) = { ϕ (y) : y ∈ Y } covers X. In this paper, we mainly show that (1) A dually CCC first countable Hausdorff space has cardinality at most 2 c and a dually weakly Lindelöf first countable normal space has cardinality at most 2 c. (2) Let Y = ∏ { Y i : i ≤ n } , where Y i is a scattered monotonically normal space for any i = 0 , 1 ,... , n. If a subspace X ⊂ Y is dually CCC then e (X) ≤ ω and a normal subspace X ⊂ Y is DCCC if and only if e (X) ≤ ω. (3) Assume 2 < c = c. A normal dually CCC space X with χ (X) ≤ c has extent at most c. (4) A dually separable Hausdorff space X with a G δ ⁎ -diagonal has extent at most c and a dually separable regular space X with a G δ -diagonal has cardinality at most c. (5) A dually CCC Hausdorff space with a G δ -diagonal has cellularity at most c. [ABSTRACT FROM AUTHOR]

Bal and Kočinac in [2] introduced and studied the class of selectively star-ccc spaces. A space X is called selectively star-ccc if for every open cover U of X and every sequence (A n : n ∈ ω) of maximal pairwise disjoint open families in X there exists a sequence (A n : n ∈ ω) such that A n ∈ A n for every n ∈ ω and St (⋃ n ∈ ω A n , U) = X. In this paper, we prove that: (1) A selectively star-ccc space is DCCC and a selectively star-ccc perfect space is CCC. (2) There exists a pseudocompact (hence, DCCC) space that is not selectively star-ccc. (3) Every selectively star-ccc subspace of the product of finitely many scattered monotonically normal spaces has countable extent. (4) Every selectively star-ccc subspace of ω 1 ω has countable extent. (5) Under 2 ℵ 0 = 2 ℵ 1 , there exists a selectively star-ccc normal space having a regular closed G δ -subset which is not selectively star-ccc. (6) Every first countable selectively star-ccc space with a G δ -diagonal has cardinality at most c. (7) Every selectively star-ccc space with a rank 2-diagonal has cardinality at most c. [ABSTRACT FROM AUTHOR]