Structural Properties of Fuzzy β-Continuous and M-Fuzzy β-Continuous Mappings

This paper establishes several results on fuzzy β-continuous and M-fuzzy β-continuous mappings between fuzzy topological spaces. Equivalent characterizations of fuzzy β-continuous maps are obtained. In particular, it is proved that a mapping is fuzzy β-continuous if and only if the inverse image of every fuzzy closed set is fuzzy β-closed. Additional characterizations are derived using β-closure operators, where necessary and sufficient conditions of the form βcl(f⁻¹(V)) ≤ f⁻¹(βcl(V)) are established for arbitrary fuzzy sets. The central contribution of this work concerns the composition of fuzzy β-continuous mappings. It is shown that the composition of two fuzzy β-continuous mappings need not be fuzzy β-continuous in general; however, sufficient conditions are provided under which the composition becomes fuzzy β-continuous, particularly when one of the mappings satisfies stronger continuity conditions such as fuzzy pre-continuity or fuzzy semi-continuity. Several propositions formalize these results, and explicit examples illustrate that the corresponding converse statements fail. This theorem clarifies the structural behavior of fuzzy β-continuity under composition and provides useful criteria for analyzing the stability of such mappings in fuzzy topological spaces. The relationships between fuzzy β-continuous mappings and other classes of fuzzy mappings are also examined. It is proved that every fuzzy continuous, fuzzy pre-continuous, and fuzzy semi-continuous mapping is fuzzy β-continuous, while the reverse implications do not hold in general. The notion of M-fuzzy β-continuous mappings is studied, and it is shown that every M-fuzzy β-continuous map is fuzzy β-continuous, although the converse fails. Finally, for a bijective mapping between fuzzy topological spaces, it is proved that if βint(A) ≤ f⁻¹(int(f(A))) for any fuzzy set A, then f(βcl(A)) ≤ cl(f(A)).