Spheres and balls in p-adic topology

Question

I'm studying about p-adic numbers, p-adic integers and p-adic topology.

I have seen:

$S(0,1)= (1 + p\mathbb{Z}_{p}) \cup (2+ p\mathbb{Z}_{p}) \cup ... \cup ( (p-1) + p\mathbb{Z}_{p}) $

when $s(a,r)= \lbrace x \in \mathbb{Q}_{p} : \vert x-a \vert_{p}=r \rbrace$

in the book p-adic Analysis Compared with Real by Svetlana Katok. In page 58, there is a figure for $ \mathbb{Z} _{5}$. I don't understand this figure.

And now, can we write something like above for other spheres?

Forexample, $S(0,\frac{1}{25})= (1 + 25\mathbb{Z}_{p}) \cup (2+ 25\mathbb{Z}_{p}) \cup ... \cup ( 24 + 25\mathbb{Z}_{p}) $ in $\mathbb{Z}_{5}$ is correct?

Answer 1

Remember that two balls in an ultrametric space are disjoint or one is contained in the other, that's one of the basic facts about geometry of such spaces.

Let $p = 5$.

Now the sets $k + 25 \mathbb Z_p$ for $k = 1, 2, \ldots, 24$ are balls around $k$ with radius $5^{-2}$. Are they disjoint? Yes, because the difference $k_1 - k_2$ for $1 \le k_1 < k_2 \le 24$ will never be divisible by 25.

Their sum is

$$\mathbb Z_p \setminus 25 \mathbb Z_p = \{x \in \mathbb Z_p : |x|_p \le 1\} \setminus \{x \in \mathbb Z_p : |x|_p \le 5^{-2}\} = \{x \in \mathbb Z_p : |x|_p \in \{1, \frac 15\}\}$$