Lemma 1.9.
Assuming that:
(
K
,
|
∙
|
)
is a
non-archimedean
valued field
Then
(i)
If
z
∈
B
(
x
,
r
)
, then
B
(
z
,
r
)
=
B
(
x
,
r
)
– so open balls don’t have a centre.
(ii)
If
z
∈
B
¯
(
x
,
r
)
then
B
¯
(
x
,
r
)
=
B
¯
(
z
,
r
)
.
(iii)
B
(
x
,
r
)
is closed.
(iv)
B
¯
(
x
,
r
)
is open.