Universal ultrametric spaces of smallest weight
by
Jerry E. Vaughan
UNC-Greensboro

A metric d on a set X is called an ultrametric provided it satisfies the following stronger form of the triangle inequality: for every x, y, z in X, d(x, y) <= max{d(x, z), d(z, y)}. The weight of a metric space is the smallest cardinality of a base for the space (which is the same as the smallest cardinality of a dense subset of the space). An isometry between two metric spaces is a one-one, onto function that preserves distances (hence is a homeomorphism). A. Lemin and V. Lemin constructed an ultrametric space LW_\tau which is universal (in the sense of isometry) for ultrametric spaces of weight at most \tau. That is, every ultrametric space of weight at most \tau can be embedded by an isometry into LW_\tau. Every such universal ultrametric space must have weight at least \tau. The Lemins raised the problem: for cardinals \tau such that \tau < \tau^\omega , is it possible to find such a universal space with weight less than \tau^\omega, and in particular, is there such a space of weight \tau? We prove that a certain subspace of LW_\tau, that we call LW_\tau', provides an affirmative solution to the Lemin's problem for every strong limit cardinal \tau of countable cofinality (hence \tau < \tau^\omega) and moreover, under the singular cardinal hypothesis, a set-theoretic assumption whose negation is related to large cardinals, the weight of LW_\tau' is \tau for all \tau greater than the cardinality of the continuum. This provided a complete, consistent solution to the problem raised by the Lemins, but their problem is still open in certain models constructed using large cardinals, e.g., a model of M. Magidor where (\aleph_\omega+1)^\omega = \aleph_\omega+2.

Date received: June 4, 1999