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Given a space X, we will say that a class of subsets of X is dominated by a class Ɓ if for any A ∈ , there exists a B ∈ Ɓ such that A ⊂. In particular, all (closed) discrete subsets of X are countably dominated (which we frequently abbreviate as ω-dominated) if, for any (closed) discrete set D ⊂ X, there exists a countable set B ⊂ X such that D ⊂. In this paper, we investigate the topological properties of spaces in which (closed) discrete subspaces are dominated either by countable subsets or by Lindelöf subspaces. [ABSTRACT FROM AUTHOR]

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]

A space X is selectively absolutely star-Lindelöf [1,3] if for each open cover U of X and any sequence ( D n : n ∈ ω ) of dense subsets of X , there are finite sets F n ⊆ D n ( n ∈ ω ) such that S t ( ⋃ n ∈ ω F n , U ) = X . In this paper, we continue to investigate topological properties of selectively absolute star-Lindelöf spaces, and show the following statements: (1) There exists a Tychonoff selectively a-star-Lindelöf, pseudocompact space X having a regular closed G δ subset which is not star-Lindelöf (hence not selectively a-star-Lindelöf); (2) Assuming 2 ℵ 0 = 2 ℵ 1 , there exists a normal selectively a-star-Lindelöf space X having a regular closed G δ subset which is not star-Lindelöf (hence not selectively a-star-Lindelöf); (3) An open F σ -subset of a selectively a-star-Lindelöf space is selectively a-star-Lindelöf; (4) For any cardinal κ , there exists a Tychonoff selectively a-star-Lindelöf, pseudocompact space X such that e ( X ) ≥ κ . [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]

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]