In this second part we continue the study of the notion of GT31 2 -spaces which we had introduced in part I of this paper. We study here its relation with the L-proximity spaces defined by Katsaras in 1979, the L-uniform spaces defined by G¨ahler and the first author and others in 1998 and the L-compact spaces defined by G¨ahler in 1995. The relation between the GT31 2 -spaces and the L-topological groups will be studied in a separate paper. Keywords: L-filters; GTi-spaces; completely regular spaces; GT31 2 -spaces; L-Tychonoff spaces; Initial and final lifts; Initial and final L-topological spaces; L-proximity spaces; L-uniform spaces; L-compact spaces; L-topological groups.