Ultrametric Space - Properties

Properties

From the above definition, one can conclude several typical properties of ultrametrics. For example, in an ultrametric space, for all and :

• Every triangle is isosceles, i.e. or or .

In the following, the concept and notation of an (open) ball is the same as in the article about metric spaces, i.e.

.

• Every point inside a ball is its center, i.e. if then .
• Intersecting balls are contained in each other, i.e. if is non-empty then either or .
• All balls are both open and closed sets in the induced topology. That is, open balls are also closed, and closed balls (replace with ) are also open.
• The set of all open balls with radius r and center in a closed ball of radius forms a partition of the latter, and the mutual distance of two distinct open balls is again equal to .

Proving these statements is an instructive exercise. Note that, by the second statement, a ball may have several center points that have non-zero distance. The intuition behind such seemingly strange effects is that, due to the strong triangle inequality, distances in ultrametrics do not add up.