Suppose $G$ is a compact topological group. We can construct the profinite completion of $G$; let's call this $\Gamma$.
My questions are:
1) Assuming that we know nothing about the (original) topology of $G$ other than that it is compact, is there anything we can say linking the topology of $G$ to the topology of $\Gamma$?
I assume the answer to this question is "no" since it seems to me that we usually regard $G$ as an abstract group when thinking about constructing its profinite completion.
2) If not (and, like I said, I assume the answer to (1) is "no") is there a way to construct a profinite completion (or something like this) of $G$ that takes into account the topology we already have?