On The Distinctive Bi-generations That are Arising Three Frameworks for Maximal and Minimal Bitopologies spaces, Their Relationship to Bitopological Spaces, and Their Respective Applications

Abstract

Due to the widespread use of various mathematical concepts, operations, relations, findings, numerous writers have established these concepts in minimal spaces. Determining how to create pairwise minimal spaces by utilizing a variety of set operators is what this article is about. Specific kinds of minimal spaces and their classical bitopologies interact with one another to form symmetry. Through the study of sets, we can investigate the characteristics and behaviors of traditional bitopological ideas. Closed spaces are a new class of bitopological spaces that we characterize and assess in this study. We also establish links among this novel category of minimal spaces and other classes of generalized spaces. Furthermore, we introduce and analyze the closed spaces that were originally suggested here, illustrate this innovative idea, make clear the interactions that go along with it, pinpoint the requirements for its successful application, and offer applications and counter-examples. Additional explanations are provided for the pairwise minimal Hausdorff Spaces, pairwise minimal Lindelöf closed spaces, pairwise minimal compact closed spaces, and maximal and minimal bitopologies. With of revenue of these spaces, we look at inverse images having particular bitopological characteristics. The discussion concludes with the identification of related product conclusions for these ideas.