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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]

In this paper, we prove that the cardinality of a space X with a symmetric g -function such that ∩ { g 2 ( n , x ) : n ∈ ω } = { x } is at most c if X satisfies one of the following conditions: (1) X has countable chain condition; (2) X is star countable (even star σ -compact); (3) X is DCCC (defined below) and normal space. We also prove that if X is a DCCC space with a symmetric g -function such that ∩ { g 3 ( n , x ) : n ∈ ω } = { x } then the cardinality of X is at most c . Finally, we make some observations on Moore spaces. [ABSTRACT FROM AUTHOR]

We prove that if a space $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ with a rank 2-diagonal either has the countable chain condition or is star countable then the cardinality of $X$ is at most $\mathfrak{c}$. [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]

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 for 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 first provide some sufficient conditions for ccc spaces to be selectively 2-star-ccc, which partially answer Problem 4.4 of Bal and Kočinac [2]. We give a Tychonoff example of a pseudocompact selectively 2-star-ccc which is not strongly star-Lindelöf, which gives a positive answer to Problem 4.8 from [2] and Question 3.11 from [16]. We also show that a regular closed G δ -subspace of a Tychonoff pseudocompact selectively star-ccc space may not be selectively star-ccc. We finally prove that the product of a selectively star-ccc space and a Lindelöf space may not be selectively star-ccc. [ABSTRACT FROM AUTHOR]