SD-Separability in Topological Spaces

Abstract

Our aim is to introduce a generalization of dense set in topological space, namely SD-dense set, when we used the notion
of somewhere dense closure operator. We provide the characterization of this class of sets, and their implications with dense sets and
with somewhere dense sets, and study their union and intersection properties, moreover we discuss their behavior as subspaces in some
special cases, additionally, we investigate their properties in some particular spaces, and then we prove that SD-dense sets, dense sets,
somewhere dense sets and open sets are equivalent in strongly hyperconnected space, after that we illustrate the image of SD-dense sets
by particular maps; as SD-irresolute map and SD-continuous map. Finally, we define a new axiom of separability, namely SD-separable
space using the notion of SD-dense sets, then we state that SD-separable space is stronger than separable space, and in submaximal
space these notions become equivalent, moreover we study the subspaces and the images of SD-separable spaces.