Polish ultrametric Urysohn spaces and their isometry groups
Su Gao and Chuang Shao
Abstract
In this paper we
give some new constructions of Polish ultrametric Urysohn spaces and
investigate the universality properties of their isometry groups. It is shown
that all isometry groups of Polish ultrametric Urysohn spaces, regardless of
their distance sets, are embeddable into each other, and in particular
universal for all isometry groups of Polish ultrametric spaces. We also
consider a strengthening notion, called extensive isometric embedding, and show
that any isometric embedding from a compact ultrametric space into a Polish
ultrametric Urysohn space is extensive. It is shown that every isometry between
two compact subsets of a Polish ultrametric Urysohn space can be extended to an
isometry of the entire space. We introduce a notion of generalized trees to
study Polish ultrametric spaces and prove a duality theorem between the
categories of Polish ultrametric spaces and their generalized tree
representations. Finally we draw some conclusions about the descriptive
complexity of embeddability, biembeddability and isometry relations among
Polish ultrametric spaces.
Table of Contents
1. Introduction
2. Preliminaries
3. The point-by-point construction
4. The Vestfrid type construction
5. The Katetov type construction
5.1 Ultrametric admissible functions
5.2 Functions with finite support
5.3 The separability of ER(X,ω) and ER(X)
5.4 The Katetov type construction
6. The generalized tree construction
6.1. Branches of R-trees
6.2. The duality theorem
6.3. Extensive isometric embeddings
7. S∞-universality of isometry groups
8. Notions of classification for Polish ultrametric spaces
References