© 1996, Topology Atlas

Let X be a set and (Gamma,¾) a partially ordered set with smallest element 0. Let

• d(x,y) = 0 iff x=y,
• d(x,y) = d(y,x),
• if d(x,y) ¾ gamma and d(y,z) ¾ gamma, then d(x,z) ¾ gamma.

An important special case occurs if (Gamma,¾) is totally ordered. If Gamma is the set of non-negative reals, one obtains as a special case of ultrametric spaces the non-Archimedean metric spaces.

The theory of ultrametric spaces dates back to the study of these spaces initiated by M. Krasner and F. Monna in the late forties; whereas a general theory of ultrametric spaces in the sense as defined above is quite new.

The most important examples of ultrametric spaces are given by valued (or semivalued) fields and groups. Boolean and Brouwerian lattices represent another class of important examples. Thus by studying ultrametric spaces, it is possible to study common properties of all these different algebraic structures, e.g., completion or embedding theorems. For every spherically complete (a strengthening of complete) ultrametric space there holds a Banch-like fixed point theorem. This theorem could also be of importance for logic programming.

Ultrametric spaces are also closely related to the so-called "metric spaces" which were studied in E. M. Jawhari, D. Misane and M. Pouzet, Retracts: graphs and ordered sets from the metric point of view, in: Combinatorics and ordered sets (Arcata, Calif., 1985), 175-226, Contemp. Math., 57 (1986).

Obviously, there is also a connection to topology. An ultrametric space canonically induces a zero-dimensional topology provided that Gamma - {0} is downward directed and has no least element. If Gamma is even linearly ordered one obtains exactly the non-Archimedean and omega_µ-metric spaces (for µ > 0). On the other hand, by non-trivial results, there exist even non-normal ultrametric spaces.