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A space X is almost star σ-compact (weakly star σ-compact) if for each open cover of X, there exists a σ-compact subset F of X such that. In this paper, we investigate the relationships among star σ-compact spaces, almost star σ-compact spaces and weakly star σ-compact spaces, and also study topological properties of almost star σ-compact spaces. [ABSTRACT FROM AUTHOR]

A space X is said to be cellular-Lindelöf if, for every family U of disjoint non-empty open sets of X, there is a Lindelöf subspace L ⊂ X , such that U ∩ L ≠ ∅ for every U ∈ U . This class of spaces was introduced by Bella and Spadaro in 2007. In this paper, our main result is to show that the Pixley–Roy space F [ X ] is cellular-Lindelöf if and only if it is CCC. We also establish a cardinal inequality for cellular-Lindelöf spaces which have a symmetric g-function. Some open questions are posed. [ABSTRACT FROM AUTHOR]

In this paper, we prove that if X is a space with a regular Gδ-diagonal and X2 is star Lindelöf then the cardinality of X is at most 2c. We also prove that if X is a star Lindelöf space with a symmetric g-function such that ∩ {g2(n, x): n ∈ ω} = {x} for each x ∈ X then the cardinality of X is at most 2c. Moreover, we prove that if X is a star Lindelöf Hausdorff space satisfying Hψ(X) = κ then e(X) ≦ 22κ; and if X is Hausdorff and we(X) = Hψ(X) = κsubset of a space then e(X) ≦ 2κ. Finally, we prove that under V = L if X is a first countable DCCC normal space then X has countable extent; and under MA+¬CH there is an example of a first countable, DCCC and normal space which is not star countable extent. This gives an answer to the Question 3.10 in Spaces with property (DC(ω1)), Comment. Math. Univ. Carolin., 58(1) (2017), 131-135. [ABSTRACT FROM AUTHOR]

A space X is said to be star determined by compact metrizable subspaces (star-CM for short) if for any open cover U of X there is a compact and metrizable subspace Y ⊂ X such that St (Y , U) = X . This notation of star-CM was introduced by van Mill, Tkachuk and Wilson in (Topol Appl 154:2127–2134, 2007). In this paper, we investigate the relations between star-CM spaces and related spaces, and study topological properties of star-CM spaces. We also establish a cardinal theorem for star-CM spaces with symmetric g-functions. [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 show that there exists a Tychonoff selectively 2-star-ccc space which is neither strongly star Lindelöf nor selectively star-ccc, which gives a positive answer to a question of Bal and Kočinac [2]. Under 2 ℵ 0 = 2 ℵ 1 , we even provide a normal example of a selectively 2-star-ccc space which is neither strongly star Lindelöf nor selectively star-ccc. Finally, we prove that every open F σ -subset of a selectively star-ccc space is selectively star-ccc. Some new questions are also posed. [ABSTRACT FROM AUTHOR]

The class of weakly linearly Lindelöf spaces was introduced and studied by Juhász, Tkachuk and Wilson in [7]. Recall that a space X is weakly linearly Lindelöf if for any family U of non-empty open subsets of X of regular uncountable cardinality κ , there exists a point x ∈ X such that every neighborhood of x meets κ -many elements of U. In this paper, we show that: (1) If X is a weakly linearly Lindelöf space and U is an open cover of X , then for the open cover { St 2 (x , U) : x ∈ X } of X , there exists a countable subset A ⊂ X such that St 2 (A , U) ‾ = X ; (2) Every weakly linearly Lindelöf normal metaLindelöf space is weakly Lindelöf; (3) If X is a first countable regular space, then M (X) (generated by Moore Machine) is weakly linearly Lindelöf if and only if X is weakly linearly Lindelöf; (4) Every product of a weakly linearly Lindelöf space and a space of countable spread (or a separable space) is weakly linearly Lindelöf; (5) If a subspace X ⊂ ω 1 ω is weakly linearly Lindelöf, then X is second countable (and hence, metrizable); (6) If X is a weakly linearly Lindelöf Baire space with a rank 2-diagonal such that w e (X) ≤ ω 1 , then | X | ≤ c ; (7) The space X is cellular-WLL if and only if it is weakly linearly Lindelöf. [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]

The class of cellular-Lindelöf spaces was introduced by A. Bella and S. Spadaro (2017) [5]. Recall 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. Cellular-Lindelöf spaces generalize both Lindelöf spaces and spaces with the countable chain condition. In this paper, we first discuss some basic properties of cellular-Lindelöf spaces such as the behavior with respect to products and subspaces. We also establish cardinal inequalities for cellular-Lindelöf quasitopological groups by using Erdös-Radó's theorem. Finally, we introduce and study the class of cellular-compact (cellular- σ -compact) spaces. In particular, we prove that every cellular- σ -compact Hausdorff space having either a rank 2-diagonal or a regular G δ -diagonal has cardinality at most c , which partially answers Question 8 and Question 9 of S. Spadaro and A. Bella (2018) [6]. Some new questions are also posed. [ABSTRACT FROM AUTHOR]