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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]

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]