A Study on b (i,j) I-Open Set within Ideal Biopological Spaces

Abstract

This paper introduces the concepts of  -open sets and -closed sets in the framework of ideal ditopological spaces, presenting them as a new form of generalized open and closed sets. -open sets are defined as subsets satisfying specific generalized openness conditions with respect to two topologies and an ideal, while  -closed sets are their complements. The study begins with a precise definition of these sets and explains how they extend previously developed ideas in generalized and bitopological structures. It examines their relationships with other known types of open and closed sets and highlights essential properties that characterize their behavior. Particular attention is given to how the interaction of the two topologies and the ideal shapes their structure and flexibility. The paper also introduces the related operators,  -interior and  -closure, and discusses their key properties. Overall, the study contributes to understanding openness and closedness in ideal bitopological spaces and provides a foundation for further research.

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