Symmetrically Fuzzy β-Connected and Disconnected Sets in Fuzzy Topological Spaces

In fuzzy topology, symmetry plays a central role in analyzing generalized forms of connectedness. This study investigates the symmetric properties associated with fuzzy β-connectedness and fuzzy β-disconnectedness in fuzzy topological spaces. Let (X, τ) be a fuzzy topological space. Two non-empty fuzzy sets A and B in X are called fuzzy β-separated if A ∧ βcl(B) = 0_X and βcl(A) ∧ B = 0_X. A fuzzy set A ≤ X is called symmetrically fuzzy β-disconnected if both A and its complement A^c can be expressed as unions of two fuzzy β-separated sets; otherwise, A is symmetrically fuzzy β-connected. Several symmetry-preserving properties are established: the complement of a symmetrically fuzzy β-connected set is also symmetrically fuzzy β-connected, and the union of two symmetrically fuzzy β-connected sets that are not completely disjoint remains symmetrically fuzzy β-connected. Furthermore, if f: X → Y is an M-fuzzy β-continuous map satisfying f⁻¹(βcl_Y(B_i)) = βcl_X(f⁻¹(B_i)) for appropriate fuzzy sets B_i, then the preimage of a symmetrically fuzzy β-disconnected set is also symmetrically fuzzy β-disconnected, demonstrating that symmetry is preserved under fuzzy continuous mappings. In general, the intersection of two symmetrically fuzzy β-connected sets is not guaranteed to be symmetrically fuzzy β-connected; however, it is preserved when one set is contained in the other. These findings highlight the fundamental importance of symmetry in understanding and analyzing fuzzy β-connectedness, providing a rigorous framework for symmetric decomposition, connectivity analysis, and the structural characterization of fuzzy sets. Overall, this study contributes to a deeper comprehension of the interplay between symmetry and connectivity in fuzzy topological spaces, offering valuable theoretical tools for further research and applications in fuzzy mathematics.