The idea is that all topologies on G (not necessarily making it a topo group) can be completely specified by a set of functions $F = \{f: G \to G\}$ if you form a basis for the topology like: $B = \{U\subset G : U = \{f(g): g \in G\}, f \in F\}$.
Now deduce properties of $F$, for instance since $aU$ must be open for all $a \in G$ we have that $F$ is closed under left or right mult.-type actions by $G$. One good reason to do this with basis $B$ and not entire topology $T$ is that $T$ is closed under union but how do you union two functions both coming from $G$?, for instance what if $f_1(g) = a \neq b = f_2(g)$ i.e. at the same $g\in G$. So there's one problem I don't know how to solve yet. Any takers?
Anyhow, I think I got it right by using basis or sub-basis as in this article:
Three interesting $\Bbb{Z}$ topologies.
The main part I'm stuck on is coming up with a characterization for $F$ when $(G, \tau(B))$ forms a topological group.
What topological group theory could I apply to this problem?