balls in ultrametric spaces are clopen subsets

In an ultrametric space, both open and closed ballsPlanetmathPlanetmath are clopen subsets.

It is indeed straightforward (exercise!) to show that the set of all open balls of radius r, centered in any of the points of a closed ball of radius r, forms a partitionMathworldPlanetmathPlanetmath 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