A study on R closed sets in topological spaces

Abstract

newline Abstract newlineThe word topology denotes the mathematical subject and the newlinename of a mathematical structure. A topology on a set is a collection newlineof open subsets satisfying certain associations about their intersections, newlineunions and complements. newlineIn this work, a new class of closed sets namely R#-closed sets newlineare defined in topological spaces by using generalized closed sets and R* newlineopen sets. Also, study the properties on R#-closed sets in topological newlinespaces. We shows that union of two R#-closed sets is R#-closed set but newlinenot their intersection. The stronger, weaker and independent form of newlineR#-closed sets with various closed sets in topological spaces are established newlineand shown in the figure. The complement of R#-closed namely newlineR#-open sets have been defined. Using the basic concepts of R#-closed newlineand R#-open sets, R#-interior, R#-closure, R#-neighbourhood and R#- newlinelimit points have been introduced in topological spaces and basic properties newlinerelated to them are discussed. newlineAlso, introduced and studied R#-continuous functions in topological newlinespaces and their different kinds as mentioned in chapter 3. The figure newlinehas been drawn to show the relation between some existing continuous newlinefunctions and R#-continuous functions. Also, a new type of closed maps newlinenamely R#-closed maps and R#-open maps are introduced and studied newlinevii newlinein topological spaces. The relation between R#-closed map and various newlineclosed maps in topological spaces is shown in the figure. Also, the basic newlineproperties of R#-homeomorphisms are studied. newlineAlso studied and investigated a type of separation axioms namely newlineR# and#1048576; Tk spaces (k = 0; 1; 2) and basic properties of these separation axioms newlineare discussed. Further, R#-regular space and R#-normal spaces have newlinebeen introduced and investigated in topological spaces. newlineFurther, We introduced new locally closed sets called R#-locally newlineclosed sets, R#locally star closed sets, R#-locally double star closed sets newlinein topological sapces and they are found to be generalization of R#- newlineclosed sets in topological spaces. Also functio