An Extensive Study on the Topological Structure of Multiple Set

Multiple sets is a recently developed mathematical framework designed to manage uncertainty and multiplicity simultaneously. They are characterized by membership matrices, which allow them to represent multiple uncertain features of objects and their corresponding multiplicities. This paper presents an in-depth study of the topological structure of multiple sets, extending existing theories of basis, interior and closure in a multiple topological spaces (MTS). We introduce the notions of subbasis, local basis, C1 space and C11 spaces, neighbourhoods, limit points, derived sets, compactness, multiple closure spaces (MCS), sequences of multiple sets and M-continuous functions within multiple topological space (MTS). Several results related to these concepts have been proven. Additionally, we provide illustrative examples of multiple topological space (MTS) and analyze their key characteristics.