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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]

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 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 spaceXismonotonically star Lindelöfif one assign to for each open covera subspace, called a kernel, such that s() is a Lindelöf subset ofX, and, and ifrefinesthen, where. In this paper, we investigate the relationship between monotonically star Lindelöf spaces and related spaces, and also study topological properties of monotonically star Lindelöf spaces. [ABSTRACT FROM PUBLISHER]

Abstract: In this paper, we continue to study the versions of property (a), and show the following statements: [(1)] There exists a Tychonoff countably compact space X such that is selectively (a), but X is not selectively (a); [(2)] There exists a Tychonoff selectively (a)-space X having a regular-closed subspace which is not selectively (a); [(3)] Assuming , there exists a normal space X that is not selectively (a); [(4)] If X is a normal selectively (a)-space with , then is selectively (a). The statement (1) answers negatively Caserta, Di Maio and Kočinac (2011) [1, Question 2.11]. [Copyright &y& Elsevier]

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]