Motivation and references for 'the topology of uniform convergence'

Question

I'm a graduate student working on $\textbf{algebraic number theory}$.

While reading papers, I have seen authors mentioning that the group of characters is equipped with the topology of uniform convergence.

I tried to understand the definition of the topology of uniform convergence but it seemed to be out of scope of the introductory course on general topology. I mean it seem to be using the concept of $\textbf{uniform space}$. And I don't have enough time to read books like Bourbaki or Stephen Willard.

So could you please recommend me short exposition on the topology of uniform convergence? or explain to me what is the point of this concept?

Thank you very much.

Answer 1

Characters are functions into $\mathbb{C}.$ One can define a metric on characters as $d(\chi, \rho) = \sup_{g\in G} |\chi(g) - \rho(g)|.$ This induces a topology. This is called the topology of uniform convergence, since $\chi_n\rightarrow\chi$ in this topology if and only if $\chi_n$ converges uniformly to $\chi.$ This is an incredibly important topology to put on a space of functions.