Uniformity on generalized topological spaces - Dipankar Dey; Dhananjay Mandal; Manabendra Nath Mukherjee (Arab Journal of Mathematical Sciences, 2022)
Purpose – The present article deals with the initiation and study of a uniformity like notion, captioned μ-uniformity, in the context of a generalized topological space. Design/methodology/approach – The existence of uniformity for a completely regular topological space is well-known, and the interrelation of this structure with a proximity is also well-studied. Using this idea, a structure on generalized topological space has been developed, to establish the same type of compatibility in the corresponding frameworks. Findings – It is proved, among other things, that a μ-uniformity on a non-empty set X always induces a generalized topology on X, which is μ-completely regular too. In the last theorem of the paper, the authors develop a relation between μ-proximity and μ-uniformity by showing that every μ-uniformity generates a μ-proximity, both giving the same generalized topology on the underlying set. Originality/value – It is an original work influenced by the previous works that have been done on generalized topological spaces. A kind of generalization has been done in this article, that has produced an intermediate structure to the already known generalized topological spaces.