A Generalization for Somewhere Dense Sets with Some Applications

In this paper, we present a new generalization for somewhere dense of a topological space (Z, τ ), namely ωSWD-open subsets. We introduce the concept of this family and discuss some of their properties with the help of illustrative example. Moreover, we will show if the space (Z, τ ) is anti-locally countable and τ is finer than the cocountable topology then the class of ωSWD-open and somewhere dense subsets of (Z, τ ) will be equivalent. Moreover, we present more properties for the class of somewhere dense subsets of (Z, τ ), the most important of which is a generalization for a theorem in [1]. Furthermore, we finish this work by shedding light on one type of covering properties where we study the notion of almost ωSWD-compact spaces with some of their properties.