There are a great many posts about the $p$-adic topology, but I can never quite find the info that I'm looking for. There is much about the topology that eludes me and I have a number of questions. The text I have been reading is Lazlo Fuch's $\textit{Abelian Groups Vol 1 & 2}$ , so I will be using his notation.
Let $A$ be an Abelian group and $p$ a fixed prime. We can gather a base of neighborhoods about 0 by considering the family of subgroups $\{ p^k A \}_{k \geq 0}$. So for any $a \in A$, a neighborhood about $a$ is $a +p^k A$.
A subset is open if it is a union of sets of the form $a + p^kA$ for various $k$'s.
Question 1 If $a$ is a fixed element, then what is $(a + p^jA) \cup (a + p^kA)$?
It seems to me that the union is $a + p^kA$ assuming w.l.o.g. $j < k$
Question 1' If $a,b \in A$, then $(a + p^jA) \cup (b + p^kA)$ is...?
I'm going to guess $a+b+p^kA$ where $j < k$ but I am not sure. This guess speaks to my biggest issue concerning unions of the sets in this topology.
Question 2 What does a closed set look like? And what are some examples?
To start answering this on my own, I am going to try to describe open balls in the $p$-adic topology and then describe their complement.
So $\mid \mid a \mid \mid_p = \frac{1}{p^{h(a)}}$ where $h(a)$ is the $p$-height of $a$, i.e., $h(a) = n$ if $p^nx = a$ for some $x$ in the group. Thus for $\epsilon > 0$, an open ball of radius $\epsilon$ can be written as $$\delta (a,b) = \{ b \in A \ \mid \ \mid \mid a - b \mid \mid_p < \epsilon \}$$
For $b \in \delta (a,b)$, suppose $\mid \mid a - b \mid \mid_p = \frac{1}{p^n} < \epsilon$ where $n$ is some natural number.
Then $$a - b = p^nx \textrm{ and } b = a + p^n (-x)$$
and so $b \in a + p^nA$ if and only if $b \in \delta (a,b)$. Is this correct?
Finally getting to closed sets, the complement of $\delta (a,b)$ would be a closed set in the $p$-adic topology...but what does it look like?
$$ \delta (a,b)^c =^? \{ b \in A \ \mid \ b \in a + p^{n+j}A, j \geq 1 \}$$
That is what I am thinking...but what confuses me is that it seems like what I just described would be a union of open sets from the base of neighborhoods that I started with.
I would also think that $p^\omega A$, the set of elements of infinite $p$-height, would be a closed set in the topology...
Question 3 Exercise 1 in Fuchs Vol 1 Section 7 says that `$A[n]$ is a closed set in any topological group.'
To begin, $A[n] = \{ a \in A \ \mid \ na = 0 \}$. Now the complement of $A[n]$ would describe $\{ a \in A \ \mid \ na \neq 0 \} = \bigcup (a + p^kA)$ where $p^k | n$ which is open since it is a union of open sets. Is this thinking correct?
Thanks for powering through this (no pun intended). Any help or references are appreciated.