Bi*-topological space

Bi-topological spaces provide a general framework for studying mathematical structures endowed with two distinct topologies, enabling the investigation of concepts that cannot be fully characterized in classical topology. In this paper, we introduce a new class of bi-topological spaces, called Bi*-topological spaces, constructed from the families of semi-open and pre-open sets associated with a given topological space. The proposed structure extends several existing concepts in bi-topology and offers a broader setting for analyzing generalized topological properties. Fundamental definitions of Bi*-topological spaces are established, followed by the introduction of Bi*-open and Bi*-closed sets, interior and closure operators, derived sets, boundary, frontier, exterior, and convergence. In addition, several separation axioms and discrete properties are investigated, and relationships between Bi*-topological spaces, classical bi-topological spaces, and bi-supra topological spaces are examined. Numerous theorems are proved to demonstrate the basic characteristics of the new space, while illustrative examples are presented to clarify the independence of the introduced concepts and the non-equivalence of several converse statements. The obtained results show that Bi*-topological spaces constitute a meaningful extension of existing bi-topological structures and provide a useful framework for further theoretical developments in generalized topology and related mathematical research.