More on μ-semi-Lindelöf sets in μ-spaces

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⊂XA\subset B\subset X, where AA is μB{\mu }_{B}-semi-Lindelöf, then AA need not be μ\mu -semi-Lindelöf. We also introduce and study a new type of generalized open sets in GTSs, called ωμ{\omega }_{\mu }-semi-open sets, and investigate them to obtain new properties and characterizations of μ\mu -semi-Lindelöf sets in μ\mu -spaces.

Keywords

• generalized topology
• μ-space
• μ-open
• μ-semi-open
• μ-lindelöf set
• μ-lindelöf space
• μ-semi-lindelöf set
• μ-semi-lindelöf space
• 54a05
• 54a10
• 54d20