closed ball

English

Noun

closed ball (plural closed balls)

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