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Let P be a topological property. A space X is said to be star P if whenever is an open cover of X, there exists a subspace A ⊆ X with property P such that , where. In this paper, we investigate the relationships among star Lindelöf spaces, star almost Lindelöf spaces and star weakly Lindelöf spaces, and also study topological properties of star weakly Lindelöf spaces. [ABSTRACT FROM AUTHOR]

In this paper, we show the following statements: (1) There exists a pseudocompact star Lindelöf Tychonoff space which is not star σ-compact. (2) There exists a Tychonoff pseudocompact star countable (hence, star Lindelöf) space having a pseudocompact, Gδ regular closed subspace which is not star Lindelöf. (3) Assuming , there exists a normal star countable (hence, star Lindelöf) space having a Gδ regular closed subspace which is not star Lindelöf. (4) Let X be a space, then A(X) is star Lindelöf if and only if e(X) ≤ ω. The statement (1) gives a negative answer to Song [13, Remark 2.3]. [ABSTRACT FROM AUTHOR]

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]

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 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]

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]

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]

In this short note, we prove that if X is a first countable DCCC space with a regular G δ -diagonal, then X has cardinality at most c . [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]

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]

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]

In this paper, we prove that the cardinality of a star Lindelöf space X does not exceed c if X satisfies one of the following conditions: (1) X has a rank 3-diagonal; (2) X is normal and has a rank 2-diagonal; (3) X is first countable, normal and has a G δ -diagonal. Moreover, we also obtain several results concerning the general question “When must a star Lindelöf space be star countable?”. [ABSTRACT FROM AUTHOR]

Kočinac and Singh [4] introduced and studied the class of set selectively k-star-ccc spaces. Let k ∈ N , a space X is said to be set selectively k-star-ccc if for each nonempty subset B of X and for every collection U of open sets in X such that B ‾ ⊂ ∪ U and for every sequence (A n : n ∈ N) of maximal cellular open families in X , there is a sequence (A n : n ∈ N) such that for every n ∈ N , A n ∈ A n and B ⊆ S t k (⋃ n ∈ N A n , U). In this paper, we show that for k ∈ N , there exists a Tychonoff pseudocompact selectively 2-star-ccc space X which is not set selectively k-star-ccc, which gives an answer to the Problem 5.1 of [4] , and construct an example of a Tychonoff set selectively star-ccc space having a regular closed G δ -subspace which is not set selectively star-ccc, which gives an answer to the Problem 5.3 of [4]. We also prove that every open F σ -subset of a set selectively star-ccc space is set selectively star-ccc, which gives a positive answer to the Problem 5.6 of [4]. Finally, we obtain a normal example of a selectively 2-star-ccc space X which is neither set strongly star-Lindelöf nor set selectively star-ccc under some set-theoretic assumption, which gives a consistent positive answer to Problem 5.4 of [4]. [ABSTRACT FROM AUTHOR]

We prove that if $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}X$ is a space satisfying the discrete countable chain condition with a rank 3-diagonal then the cardinality of $X$ is at most $\mathfrak{c}$. [ABSTRACT FROM PUBLISHER]

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]

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]

In this paper, we make several observations on quasitopological groups, covering properties and cardinal inequalities. In particular, we show that: (1) For every quasitopological group G , i b (G) ≤ e (G) ; (2) If G is a T 1 -quasitopological group then G has cardinality at most 2 e (G) ψ (G) , which partially answers a question asked by Arhangel'skii and Tkachenko in [3] and a question asked by Tkachenko in [20] ; (3) In the class of first countable quasi-topological groups, dually separable is self-dual with respect to neighborhood assignments; (4) If X is a Baire weakly star countable space with a rank 2-diagonal then X has cardinality at most 2 ω , which gives a partial answer to a question of Gotchev in [11] ; (5) If X is a quasiLindelöf space with a rank 2-diagonal then X has cardinality at most 2 ω ; (6) In any model of ZFC containing a 2 ω -Suslin Line, there exists a Moore space of cellularity at most 2 ω which has cardinality greater than 2 ω , which answers a question of Arhangel'skii and Bella in [2]. Some new questions are also posed. [ABSTRACT FROM AUTHOR]

In this paper, we study some new selection principles using the star operator which was introduced by Bal and Bhowmik (2017) [1]. We first prove that there exists a space X which has the property U f i n ⁎ (O , O) but does not have the property U f i n ⁎ (O , O) , where O denotes the collection of all open covers of a space X. We also obtain several examples of spaces having the property U 1 ⁎ (O , O) but their products do not have the property U 1 ⁎ (O , O). A Tychonoff example of a space having the property U 1 ⁎ (O , O) which is not star countable is also given. If a space X has the property U 1 ⁎ (O , O) (respectively, U f i n ⁎ (O , O)) and e (X) < ω 1 , then the Alexandroff duplicate A (X) has the property U 1 ⁎ (O , O) (respectively, U f i n ⁎ (O , O)). Finally, we prove that the property U 1 ⁎ (O , O) is not hereditary with respect to regular closed subsets and every regular paraLindelöf 1-star-Lindelöf space is Lindelöf. The above-mentioned results answer two published open questions from [1]. [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]

The notion of a κ -splitting diagonal was introduced and studied by Tkachuk. In this paper, we prove that there exists a locally countable, locally compact space with an ω 1 -splitting diagonal but no G δ -diagonal. Using the Erdös-Radó's theorem, we also prove that every DCCC homogeneous space X with a regular G δ -diagonal such that π χ (X) = ω has cardinality at most c. Some new questions are also posed. [ABSTRACT FROM AUTHOR]

In this paper, we mainly consider some cardinal invariants and grasps of quasitopological groups and some properties of two classes of quasitopological groups. We show that: (1) There exists a pseudocompact quasitopological group K with countable cellularity and I n (K) > ω , which gives a negative answer to [30, Question 5.3] ([27, Question 3.6]); (2) There exists a pseudocompact quasitopological group G of countable cellularity and uncountable g -tightness, which gives a negative answer to [8, Open problem 6.4.9] ; (3) There exists a Tychonoff quasitopological group G containing compact invariant subgroups F , M such that G = F M , but the space G is not Čech-complete, which gives a partial answer to [8, Open problem 4.6.9] ; (4) A Fréchet-Urysohn quasitopological group G with sequentially continuous multiplication is a strong α 4 -space; and as an application, we give a partial answer to [9, Question 2.4]. We also introduce the concept of strong quasitopological groups. Some properties of strong quasitopological groups are obtained, which generalize some properties of topological groups. As some applications, we give some partial answers to related open problems. [ABSTRACT FROM AUTHOR]