Vague Topological Modules and Vague Topological Vector Spaces

In this paper, we introduce and develop the notions of vague topological ring, vague topological field, vague topological module, and vague topological vector space within the framework of vague set theory. By combining the algebraic structures of rings, fields, and modules with vague topology, we establish a generalised setting that extends classical topological algebra. We present fundamental definitions, illustrative examples, and examine important structural properties of these newly defined concepts. Furthermore, we investigate the relationships between vague topological structures and their corresponding classical counterparts, and analyse continuity conditions of algebraic operations under vague topology. The results obtained contribute to the advancement of vague algebraic topology and provide a foundation for further research in this direction.