closed ball

Noun

 * 1)  A 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 $$ \{x | \exists n \in \mathbb{Z} . \, x = 3 n + 1 \} $$. This closed ball partitions into exactly three smaller closed balls of radius 1/9, e.g., $$\{x | \exists n \in \mathbb{Z} . \, x = 4 + 9 n \} $$. 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.