Novel Categories of Spaces in the Frame of Generalized Fuzzy Topologies via Fuzzy $g\mu$-closed sets

Abstract

One of the known approaches to studying topological concepts is to utilize subclasses of topology, such as clopen sets and generalized closed sets. In this study, we apply the notion of fuzzy generalized $\mu$-closed sets ($Fg\mu$-closed sets) to establish and analyze novel categories of spaces, namely $Fg\mu $-regular, $Fg\mu $-normal, and $F\mu $-symmetric spaces in the frame of generalized fuzzy topology($GFT$). We investigate the fundamental properties of these classes, exploring their unique characteristics and preservation theorems under $Fg\mu$-continuous maps. We establish the interrelationships between these classes and the other separation axioms in this setting, and we demonstrated that $F\mu $-regular, $F\mu $-normal, and $F\mu $-symmetric spaces are special cases of $Fg\mu $-regular, $Fg\mu $-normal, and $F\mu$-$T_{1}$ spaces respectively. Additionally, we show that the equivalence for these cases hold when the $GFT$ is $F\mu$-$T_{\frac{1}{2}}$. The connections between these classes and their counterparts in the crisp $GT$ are studied. Finally, we discuss these classes' hereditary and topological properties, further enhancing our comprehension of their behavior and implications.