The Assessment of Various Normal spaces in DR Nano topological space

The main goal of the study is to describe and characterize new classes of normal spaces namely , and spaces and to investigate their various properties and features.I. INTRODUCTION In 2013,Lellis Thivagar introduced the concept of Nano-topological spaces[13].Nano-topology has been analysed by various authors in a diverse array of applications [8-10].The idea of ideal Nano-topological space was provided by Parimala[6].In Ideal Nano-topological space,different varieties of generealized open and closed sets are introduced[6,12].Maheswari and Rekha have introduced a new kind of ideal Nano-topological space called -Nano topological space and explored different kinds of open sets in -Nano topological space[14].In this current research, we explore the ideas of , and -spaces in -Nano topological space and examine their various characteristics properties. PreliminariesDefinition: 2.1 [14] Let be an ideal in the universal set .Let .Then the Nano right dynamic lower, upper and boundary regions are given by () = { : i () i () }, where i () = {U: () ()}and () = { : i () i } and () = () - ().Also, the generated Nano right dynamic topology (called -Nano topology) with respect is given by () = {, , (), (), ()}.The collection () = {, (), ()} forms a basis for and the Nano right dynamic topological space is denoted by (, (), ).The elements of () are called -open sets and its complements are called -closed sets.Definition 2.2 [14] Let (, (), ) be a nano right dynamic topological space.An operator : () () is known as the dynamic function of on and defined as = { : ()} and its closure operator is * () = .Definition 2.2 [14] A subset of (, (), ) is -open if = ( () Definition 2.3 [14] Let be a subset of (, (), ).Then, is) whenever , is -open.Example 2.4.Let = {, , , , }, = {, }, = {, {}, {}, {, }} and the relation X = {(, ), (, ), (, ), (, ), (, ), (, ), (, ), (, )}.Then the corresponding -Nano topology is () = {, , {}, {, , }, {, }}.Then () = {, , {, }, {, , }, {, , , }}, * () = {, , {},