Let $G$ and $G'$ be two topological abelian groups with same underlying abelian groups. If $U$ gives fundamental system of $0$ of $G$, and $U$ also gives fundamental system of $0$ of $G'$, then, does $G$ and $G'$ has the same topology ? 　My try: Basis of both are written as $\bigcup_{g∈G}{gG}$ ...