balls in ultrametric spaces are clopen subsets
In an ultrametric space, both open and closed balls are clopen subsets.
It is indeed straightforward (exercise!) to show that the set of all open balls of radius ,
centered in any of the points of a closed ball of radius , forms a partition
of the latter.
Thus, in particular, any point of a closed ball is an interior point, and the same holds for the complement of an open ball.
| Title | balls in ultrametric spaces are clopen subsets |
|---|---|
| Canonical name | BallsInUltrametricSpacesAreClopenSubsets |
| Date of creation | 2013-03-22 18:20:15 |
| Last modified on | 2013-03-22 18:20:15 |
| Owner | MFH (21412) |
| Last modified by | MFH (21412) |
| Numerical id | 5 |
| Author | MFH (21412) |
| Entry type | Example |
| Classification | msc 54D05 |