Somewhere dense on upper bounded \(T_{0}\)-Alexandroff spaces

Abstract

Studying different types of generalized open sets is a major area of research in topology during the last few decades. In this paper, we study somewhere dense sets and\( swd\)-continuity on upper-bounded \(T_{0}\)-Alexandroff spaces. We introduce and investigate various types of generalized somewhere dense; namely \(^{*}sw\)-dense, \(sw^{*}\)-dense, and \(sw^{**}\)-dense. We prove that a subset \(A\) of a topological space is preclosed if and only if it is \(p^{*}gp\)-closed sets. We give a characterization for a set $A$ to be $pre^{*}$-closed.