I'm taking a topology course right now, and a lot of people are (understandably) confused by the profinite topology (and pro-$\mathcal{C}$ topologies more generally) on a group. The definition is a bit unwieldy, and a lot of the students taking this class haven't taken the core algebra sequence yet.
I find it's easiest to learn things when I have some motivation, and while helping some people study, I wanted to say some words about why people care about the profinite topology... Unfortunately, the usual application (and the only one that I really know) is the galois theory of infinite extensions. This is entirely unhelpful, since the people I was chatting with haven't seen classical galois theory yet!
I know that $p$-adic integers are profinite, but I know relatively little number theory, so I'm not sure if there are "simple" places where the topology of $\mathbb{Z}_p$ ends up being useful. I also realized that I don't really know how the topology reflects the underlying algebra. I know that for galois theory we need to replace "subgroup" by "closed subgroup", and a group is residually $\mathcal{C}$ exactly when it's Hausdorff with the pro-$\mathcal{C}$ topology... But that's really it.
With all this said, what are the "simplest" applications of profinite (and pro-$\mathcal{C}$) topologies that you can think of? Are there ways where the topology reflects the algebra that can be explained with relatively little background? And lastly, are there good references for pro-$\mathcal{C}$ groups that might explain some of this material in more depth?
Thanks in advance! ^_^