In this paper, we study $\omega$-narrow and $\omega$-balanced topological groups and prove they may be embedded as subgroups of products of second countable (resp, first countable) topological groups. We also prove that this kind of groups are closed with respect to the most common operations, such as the taking of subgroups, arbitrary products and under continuos homomorphic images. We finally prove that the class of $\omega$-balanced topological groups is wider than the class of $\omega$-narrow topological groups.