I am trying to work out how it is that we actually work open sets on a p-adic topological space and how I would relate it to open sets in a point set topology. According wiki here: We have that open sets are of the form: $U_a(n) = \{n + \lambda p^a : \lambda \in \mathbb{Z}_p\}$ where a is a non-negative integer and $n$ is an integer in $[1, p^a]$
Now I am trying to get the intuition of what is going on here. So say we take tuples of the form $U_a(n) = (p,a,n,\lambda)$ so letting $p= 3$ we get: As $a \in \mathbb{N} n \in [1,3^a]$, and $\lambda \in [1,\mathbb{Z_3}]$ Varying first on $\lambda$, then on $n$ and then $a$:
$U_1(1) = \{(3,1,1,0),(3,1,1,1),(3,1,1,2)\} = \{1,4,7\}$ $U_1(2) = \{(3,1,2,0),(3,1,2,1),(3,1,2,2)\} = \{2,5,8\}$ $U_1(3) = \{(3,1,3,0),(3,1,3,1),(3,1,3,2)\} = \{3,6,9\}$
Then as all the terms are exhausted for $n$ we go to $a = 2 \Rightarrow n \in [1,3^2]$ $U_2(1) = \{(3,2,1,0),(3,2,1,1),(3,2,1,2)\} = \{1,10,19\}$ $U_2(2) = \{(3,2,2,0),(3,2,2,1),(3,2,2,2)\} = \{2,11,20\}$
...
$U_2(9) = \{(3,2,9,0),(3,2,9,1),(3,2,9,2)\} = \{9,18,27\}$
Then $a = 3$ and $n \in [1,3^3]$ etc.
So I see that for each $a$ we look at the columns at the above, these 3 - tuples as rows in a 3 by $n$ matrix we are covering all elements from 1 to $3^a$
So what? What do these sets of 3-tuples tell me and how is it that these $U_a(n)$ are open sets? What I am really interested in seeing how this topology (p-adic topology?) can satisfy the ultrametric.
Thanks for your insight,
Brian