Questions tagged [ultrametric]

An ultrametric space is a metric space in which the triangle inequality is strengthened to $ d ( x , z ) \le \max \lbrace d ( x , y ) , d ( y , z ) \rbrace $. Sometimes the associated metric is also called a non-Archimedean metric or super-metric.

An ultrametric space is a set of points $ M $ with an associated distance function (also called a metric)$ d : M \times M \to \mathbb R $, such that for all $ x , y , z \in M $, one has:

• Nonnegativity: $ d ( x , y ) \ge 0 $;
• Definiteness: $ d ( x , y ) = 0 $ if and only if $ x = y $;
• Symmetry: $ d ( x , y ) = d ( y , x ) $;
• Strong triangle or ultrametric inequality: $ d ( x , z ) \le \max \lbrace d ( x , y) , d ( y , z ) \rbrace $.

From the above definition, one can conclude several typical properties of ultrametrics. For example, for all $ x , y , z \in M $, at least one of the three equalities $ d ( x , y ) = d ( y , z ) $, $ d ( x , z ) = d ( y , z ) $ or $ d ( x , y ) = d ( z , x ) $ holds. That is, every triple of points in the space forms an isosceles triangle, so the whole space is an isosceles set.

Defining the (open) ball of radius $ r > 0 $ centred at $ x \in M $ as $ B ( x ; r ) := \lbrace y \in M \mid d ( x , y ) < r \rbrace $, we have the following properties:

• Every point inside a ball is its center, i.e. if $ d ( x , y ) < r $ then $ B ( x ; r ) = B ( y ; r ) $.
• Intersecting balls are contained in each other, i.e. if $ B ( x ; r ) \cap B ( y ; s ) $ is non-empty then either $ B ( x ; r ) \subseteq B ( y ; s ) $ or $ B ( y ; s ) \subseteq B ( x ; r ) $.
• All balls of strictly positive radius are both open and closed sets in the induced topology. That is, open balls are also closed, and closed balls (replace $ < $ with $ \le $) are also open.
• The set of all open balls with radius $ r $ and center in a closed ball of radius $ r > 0 $ forms a partition of the latter, and the mutual distance of two distinct open balls is (greater or) equal to $ r $.

Source: Wikipedia