Boolean Algebras Describe Ultrametric Spaces up to Uniform Euivalence
by
Alex J. Lemin
Moscow State University of Civil Engineering

Theorem. There exists a contra-variant functor \lambda: Ultram --> Bool from the category of ultrametric spaces and uniformly continuous maps Ultram onto a subcategory Bool* of the category of Boolean algebras Bool such that algebras \lambda(X) and \lambda(Y) are isomorphic iff the completions of spaces X and Y are uniformly homeomorphic.

Corollary. Functor \lambda is one-to-one over the class of complete ultrametric spaces.

A metric space (X, d) is called ultrametric iff it satisfies the strong triangle Axiom d(x, z) <= max[d(x, y), d(y, z)] . Functor \lambda can be defined as a composition \lambda = S \sigma F of a forgetful functor F (from Ultram to a category of proximity spaces Prox, [1]), the Smirnov compactification functor \sigma (from Prox to a category of compacta Comp, [4, 6]), and Marshall Stone's duality functor S (between Comp and Bool, [3, 7]). It can be also defined explicitly. We describe its properties, its relations to Euclidean geometry [2], Lebesgue measure theory, and give a lattice characterization of algebras \lambda(X) from Bool*. Recall that (non-distributive) complete, atomic, treelike, and real graduated lattices LAT* describe ultrametric spaces up to isometry, [5].

References
1. J. Lemin. Proximity on isosceles spaces, - Russian Math. Surveys, 39:1 (1984), 143-144.
2. --. Isometric imbedding of ultrametric (=non-Archimedean) spaces in Euclidean spaces, - Soviet Math Doklady 32:3 (1985), 740-744.
3. --. Ultrametric spaces and Boolean algebras, - Fifth Int. Con "Topology and App". Zagreb, 1990
4. --. The Smirnov compactification functor is one-to-one over the class of complete first countable spaces, - "Topology and its applications", 38 (1991), 201
5. --. The category of ultrametric spaces is isomorphic to the category of complete, atomic, tree-like, real graduated lattices LAT*, Älgebra universalis" (to appear)
6. Yu. M. Smirnov, On proximity spaces, - Math. Sbornik, 31 (1952), 543-574 (in Russian), A.M.S. Trans. Ser. 2, 38, p.5-35.
7. M. Stone. Applications of the theory of Boolean rings to general topology, - Trans AMS 41 (1937), 375-481

Date received: May 3, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caee-23.