Abstract
Abstract Sarsak [On μ \mu -compact sets in μ \mu -spaces, Questions Answers Gen. Topology 31 (2013), no. 1, 49–57] introduced and studied the class of μ \mu -Lindelöf sets in μ \mu -spaces. Mustafa [ μ \mu -semi compactness and μ \mu -semi Lindelöfness in generalized topological spaces, Int. J. Pure Appl. Math. 78 (2012), no. 4, 535–541] introduced and studied the class of μ \mu -semi-Lindelöf sets in generalized topological spaces (GTSs); the primary purpose of this paper is to investigate more properties and mapping properties of μ \mu -semi-Lindelöf sets in μ \mu -spaces. The class of μ \mu -semi-Lindelöf sets in μ \mu -spaces is a proper subclass of the class of μ \mu -Lindelöf sets in μ \mu -spaces. It is shown that the property of being μ \mu -semi-Lindelöf is not monotonic, that is, if ( X , μ ) \left(X,\mu ) is a μ \mu -space and A ⊂ B ⊂ X A\subset B\subset X , where A A is μ B {\mu }_{B} -semi-Lindelöf, then A A need not be μ \mu -semi-Lindelöf. We also introduce and study a new type of generalized open sets in GTSs, called ω μ {\omega }_{\mu } -semi-open sets, and investigate them to obtain new properties and characterizations of μ \mu -semi-Lindelöf sets in μ \mu -spaces.
Highlights
Introduction and preliminariesA topological space X is said to be Lindelöf [1] if every open cover of X has a countable subcover
Lindelof spaces play a vital role in the theory of general topology as a natural generalization of compact spaces
The primary purpose of this paper is to continue the study of μ-semi-Lindelöf sets in μ-spaces introduced by Mustafa in [16]
Summary
A topological space X is said to be Lindelöf [1] if every open cover of X has a countable subcover. (i) A subset A of a topological space (X, τ) is called semi-compact relative to X [32] (i) A subset A of a μ-space (X, μ) is μ-semi-Lindelöf if and only if for every family = {Fα : α ∈ Λ} of μ-semiclosed sets having the property that for every countable subfamily i of , (⋂ i) ∩ A ≠ ∅, (⋂ ) ∩ A ≠ ∅. For a subset A of a μ-space (X, μ), the following are equivalent: (i) A is μ-semi-Lindelöf; (ii) Every maximal strong filter base on X, each of whose members meets A, μσ-converges to some point of A;. For a μ-space (X, μ), the following are equivalent: (i) X is μ-semi-Lindelöf; (ii) Every maximal strong filter base on X μσ-converges to some point of X;. (iii) Every strong filter base on X μσ-accumulates at some point of X
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