How to understand the infraconnected set and affinoid?

Question

I begin to study some p-adic analysis. I find it is hard to understand the infraconnected set and affinoid. It is strange that I cannot find them at wiki and only a few book(by the same auther) discuss them, but also no simple example. Does them has other names?

1)Please give me some standard simple examples to show thet what is the infraconnected set looks like.

2)Is the infraconnected set relate with the connected set in the topology?

3)I also find a related concept called empty annulus, does it has some special meaning in the p-adic analysis?

4) The affinoid set means bounded closed infraconnected set with finitely many holes, what is the holes here?

A similar question is at here.

Answer 1

1. Basic examples of infraconnected sets include discs and annuli.

2. Infraconnected sets in $\mathbb{C}_p$ are analogous to connected subsets of $\mathbb{C}$, in the sense that no point in such a set has an empty annulus surrounding it. Of course, we have to be rather flexible with the notion of "empty annulus" in $\mathbb{C}$ for this to really make sense, but "empty topological circle in the Riemann sphere" suffices.

3. I don't know. I'd never seen the term "empty annulus" until today, but I'm not a specialist. I'd say it is a natural thing to consider if you are restricting your attention to one dimension.

4. Holes are the maximal discs that make up the complement of a closed set $X$ in the disc containing $X$ whose radius is equal to the diameter of $X$. In algebraic geometry, one obtains finitely many punctures in the affine line by inverting polynomials, and affinoids are an analytic version of that, with some finer metric structure.

In general, if you are starting in a field where there are many books by many authors, you may not want to spend too much time learning terminology that only appears in books by one author. The more universal words are likely to be more useful. On the other hand, a quick search of Google Books suggests "infraconnected" appears in books by both Escassut and Robert, so that rule doesn't apply here.