The Set Of All Balls

Proposition 1.8.2. The set of all balls in
Q
p

is countable. Proof. Let us write the center of the ball
B
γ

(a)
as
a=
a
~
+p
−γ
ξ
, where
a
~
=
k=−m

−γ−1

a
k The Set Of All Balls

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p
k

is a rational number,
ξ∈Z
p

. Then
B
γ

(a)

={x:∣x−a∣
p

≤p
γ
}={x:∣p
γ
(x−a)∣
p

≤1}
={x:∣p
γ
(x−
a
~
)+ξ∣
p

≤1}={x:∣p
γ
(x−
a
~
)∣
p

≤1}=B
γ

(
a
~
).

Thus each ball can be characterized by a pair of numbers
(
a
~
,γ)
where both numbers belong to countable sets. Therefore the set of all pairs
(
a
~
,γ)
is also countable and so is the set of all balls in
Q
p The Set Of All Balls 

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