Cayley - MacLane type theorem for ultrametric spaces
by
Vladimir Lemin
Coauthors: Alex J. Lemin

The theory of ultrumetric spaces takes its origin from the study of the following example.

EX.1. Let LR be a space of finite functions f(x): R₊ --> R equipped with the metric d(f, g) = sup{ x | f(x) =/= g(x) }.

This metric satisfies the strong triangle inequality d(f, h) <= max[d(f, g), d(g, h)]. (\Delta) Therefore (LR, d) is an ultrametric (or non-Archimedean or isosceles) space. Later on it was revealed that almost all theorems on the properties of LR depend on inequality (\Delta), but not on the specific nature of the elements of LR. It leads us to the axiomatic definition of ultrametric space [1] and to the description of topological [2], uniform [3], metric [4], categorical [4, 5] and geometric [7] properties of these spaces and groups [6]. It turned out that example 1 and even example 2 (see below) are universal.

EX.2 Let M be a poited set with a base point a, Q₊ be the set of positive rationals and LM be a set of "finite" mappings f:Q₊ --> M equipped with the metric (*) ["finite" means that f(x)=a for all x > X(f)].

THEOREM. Every isosceles space (M, d) can be isometrically embedded in LM-space (namely in the space of finite maps from Q₊ into the set M).

PROOF is similar to one from [8] where the set of positive reals R₊ is exploited instead of Q₊.

COROLLARY. For every potency \tau there exists the universal ultrametric space LW_\tau containing isometrically all ultrametric spaces of the weight <= \tau, w(LW_\tau) = | LW_\tau | = \tau^\aleph₀.

NOTE. If 2 <= \tau <= c then \tau^\aleph₀ = c. The weight of universal ultrametric space can not be smaller, even for \tau = 2. It follows immediately from the following theorem.

THEOREM. For any ultrametric space (X, d) the set of possible values of its metric V = { d(x, y) | x, y in X } has cardinality |V| <= w(X).

Certainly if one does not need the including space to be ultrametric its weight can be made much smaller. Perhaps the most wonderful property of isosceles spaces is stated in the following theorem.

THEOREM. Every ultrametric space of weight \tau can be isometrically embedded in generalized Hilbert space H_\tau. Every (n+1)-point ultrametric space can be isometrically embedded in Euclidean n-dimensional space Eⁿ, but not in E^k for k < n.

This property is far from being trivial even for finite space in case when n >= 3. See [7] for proof, corollaries and similar properties.

REFERENCES.

1. M.Krasner. Espaces ultramétriques, C.R. Acad. Sci. Paris 219 (1944), 433
2. J.de Groot. Non-Archimedean metrics in topology, Proc. A.M.S. 7(1956), 948
3, 4, 5. A.J.Lemin. On isosceles spaces, Russian Math. Surv. 39:1, 39:5(1984); 40:6(1985)
6. -, Yu.M.Smirnov. Isometry groups of (ultra-)metric spaces, ibid. 41:6(1986), 175
7. -. Isometrical embedding of isosceles spaces in Euclidean, Sov.Math.Dokl. 285:3(1985)
8. -. in "Functional analysis and its applicatoins..." M. MSU-press, 1984 (in Russian).

Date received: June 24, 1996

Copyright © 1996 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 # caaj-37.