On Cubis Sets and Cubic Topological Spaces

Abstract

In this research work, various types of cubic sets are introduced and some results based on the newlineoperations of P-union, P-intersection, R-union and R-intersection are examined and counter examples newlineare given graphically. Also the concepts like interior, closure and various types of open and closed newlinesets, base, subbase, cubic point, continuous mappings, almost-continuous mappings and open mapping newlineare introduced in P-cubic topological spaces and R-cubic topological spaces. Also many properties and newlinecharacterizations of the newly defined open sets and continuous mappings are obtained. The concept newlineof connectedness is also studied in P-cubic topological spaces. newlinei) Major Objectives: newlineand#61623; To define various types of open sets on P-Cubic open topological spaces and compare the newlineinterrelations between them. newlineand#61623; To define various types of open sets on R-Cubic open topological spaces and compare the newlineinterrelations between them. newlineand#61623; To define various types of Continuous mappings and open mappings on P-Cubic open newlinetopological spaces and to study the interrelations between them. newlineand#61623; To define various types of Continuous mappings R-Cubic open topological spaces and to study newlinethe interrelations between them. newlineand#61623; To define various types of connectedness on P-Cubic open topological spaces and to study the newlineinterrelations between them. newlineii) Hypothesis newlineand#61656; To provide graphical representation for easy understanding. newlineand#61656; To provide examples for clear understanding of defined concepts. newlineiii) Methodology newlineand#61607; Theorems and Lemmas are proved for the interrelation between the defined sets. newlineand#61607; Examples are provided to show that the reverse implications are not true in general. newlineiv) Findings newlineand#61692; Also R-union and R-intersection the of two UIECS need not be UIECS. newlineand#61692; Also R-union and R-intersection the of two LIECS need not be LIECS. newlineand#61692; P-cubic regular open set is stronger than all other P-cubic open sets defined. newlineand#61692; R-cubic regular open set is stronger than all other P-cubic open sets defined.