Take $G$ an abelian additive group. I was looking about the following problem. If $G$ is topological we know (because of the translations are homeomorphisms) that the topology is uniquely determined by the neighborhoods of the $0.$ My question is the following
If I take a countable collection of subset of $G$, $\{G_n\}$, can I define a topology such that this is a fundamental system of neighoborhood and $G$ with this topology is a topological group?
I'm interested in the case where $G=G_0\supset G_1 \ ...$ are subgroups (for example the $I$-adic topology over a ring).