I learned from several places that in defining a character of a topological group $G$, we often require it to be continuous, i.e. $\omega:G\to \mathbb{C}^{\times}$ is a continuous group homomorphism. This is particularly the case when $G$ is the $p$-adic field $\mathbb{Q}_p$ or the idele group $\mathbb{A}^{\times}_{\mathbb{Q}}$. Is there any significance to the continuity here? Besides, is here $\mathbb{C}^{\times}$ always equipped with the induced topology from $\mathbb{C}$? Or rather, what is the topology of $\mathbb{C}^{\times}$ here?
I know that in the case of $G=\mathbb{Q}_p$, $\mathbb{C}^{\times}$ is given the discrete topology. Yet I am not very clear about the general case. Will someone be kind enough to say something on this? Thank you very much!