This paper aims to introduce some new locally closed sets called binary πg-locally closed sets, bπglc^⋆-set, bπglc^(⋆⋆)-set and the relations between them are studied in a binary topological space. The concepts of bπg-submaximal space and their related properties have been introduced. Also a characterization of a bπg-submaximal space is found, and suitable examples are given.