1.NounA ball which contains its boundary, i.e., a ball which is a closed set."In the set of 3-adic numbers, the closed ball of radius 1/3 "centered" at 1 is the set
{
x
|
∃
n
∈
Z
.
x
=
3
n
+
1
}
{\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=3n+1\}}
. This closed ball partitions into exactly three smaller closed balls of radius 1/9, e.g.,
{
x
|
∃
n
∈
Z
.
x
=
4
+
9
n
}
{\displaystyle \{x|\exists n\in \mathbb {Z} .\,x=4+9n\}}
. Then each of those balls partitions into exactly 3 smaller closed balls of radius 1/27, and the sub-partitioning can be continued indefinitely, in a fractal manner."