Some topological properties of generalized $2$-normed (G2N) spaces have been studied in this article. The notion of $g$-convergence for sequences is introduced in general, and it is compared with the usual notion of convergence. It is shown that $g$-convergence is a more general idea, and under certain conditions $g$-convergence and convergence actually coincide. Using these concepts, a few fixed point theorems are developed.