Proximity structure on generalized topological spaces

Abstract

The aim of the present article is to introduce a kind of proximity structure, termed $\mu$-proximity, on a set X, which ultimately gives rise to a generalized topology on the ambient set X. An alternative description of $\mu$-proximity is given and it is shown that any generalized topology of a generalized topological space $(X, \mu )$ is always induced by a suitable $\mu$-proximity if and only if $(X, \mu )$ satisfies a type of complete regularity condition. The notion of quasi $\mu$-proximity is also introduced and the desired result that every generalized topology can be achieved from a quasi $\mu$-proximity, is proved.