Why is the topology of characters determined by the open sets containing the trivial character? Let $G$ be an abelian topological group, and let $\hat G$ denote the set of characters on $G$. Why is it true that if one has a topological basis of for the trivial character (say the topological of uniform convergence on compact sets), one has a topology for all of the characters? This seems clear since the complex circle is a homogeneous space, but I'm having trouble formalizing this for some reason.

The basis at identity yields a basis on the entire topological group, for any topological group.
Notice that for any neighbourhood $U$ of a point $g\in G$ with $G$ a topological group, $g^{-1}U$ is a neighbourhood of identity.
