Ultrametric Space - Examples

Examples

  1. The discrete metric is an ultrametric.
  2. Consider the set of words of arbitrary length (finite or infinite) over some alphabet Σ. Define the distance between two different words to be 2-n, where n is the first place at which the words differ. The resulting metric is an ultrametric.
  3. The p-adic numbers form a complete ultrametric space.
  4. If r=(rn) is a sequence of real numbers decreasing to zero, then |x|r := lim supn→∞ |xn|rn induces an ultrametric on the space of all complex sequences for which it is finite. (Note that this is not a seminorm since it lacks homogeneity. — If the rn are allowed to be zero, one should use here the rather unusual convention that 00=0.)
  5. If G is an edge-weighted undirected graph, all edge weights are positive, and d(u,v) is the weight of the minimax path between u and v (that is, the largest weight of an edge, on a path chosen to minimize this largest weight), then the vertices of the graph, with distance measured by d, form an ultrametric space, and all finite ultrametric spaces may be represented in this way.

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