Search

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]

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]

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]

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]