Foundations

Abstract

We begin by establishing some very basic and elementary notions.

Definition. A metric space (X,d) is called ultrametric if the strict triangle inequality

$$d (x, z) \le\max(d (x, y), d (y, z)) \quad \textrm{for any} \ x,y, z \in X$$

is satisfied.

Examples will be given later on.

Remark. i. If (X,d) is ultrametric then (Y,d |Y×Y), for any subset Y⊆X, is ultrametric as well.

ii. If (X ₁,d ₁),…,(X _m ,d _m ) are ultrametric spaces then the cartesian product X ₁×⋯×X _m is ultrametric with respect to

$$d ((x_1, \ldots, x_m), (y_1, \ldots, y_m)) : = \max(d_1 (x_1,y_1), \ldots, d_m (x_m, y_m)) .$$