Lipschitz functions on topometric spaces

Itaï Ben Yaacov

doi:10.4115/jla.2013.5.8

Abstract

We study functions on topometric spaces which are both (metrically) Lipschitz and (topologically) continuous, using them in contexts where, in classical topology, ordinary continuous functions are used. We study the relations of such functions with topometric versions of classical separation axioms, namely, nor- mality and complete regularity, as well as with completions of topometric spaces. We also recover a compact topometric space X from the lattice of continuous 1-Lipschitz functions on X , in analogy with the recovery of a compact topological space X from the structure of (real or complex) functions on X. 2010 Mathematics Subject Classification 54D15 (primary); 54E99, 46E05 (sec- ondary)

Highlights

Every bounded function is Lipschitz, and if α = ∞ every function is k-Lipschitz for any k
A minimal topometric space is one in which the metric and topology agree, which may be identified with its underlying metric structure
These mostly serve as first sanity checks

Summary

Normal topometric spaces and Urysohn and Tietze style results

For two topometric spaces X and Y we define CL(1)(X, Y ) to be the set of all continuous 1-Lipschitz functions from X to Y. Let Ξ be a finite c-Lipschitz S-system in a normal topometric space. In a normal topometric space X, for every finite c-Lipschitz system Ξ and c′ > c there exists a continuous c′-Lipschitz f : X → R compatible with Ξ. Let X be a normal topometric space, F, G ⊆ X closed sets, 0 < r < d(F, G). For each n the space Sn(L) of complete n-types in L is compact, so we may apply Corollary 1.7, observing that the n-ary L-definable predicates are in a natural bijection with the continuous function on Sn(L), and that this bijection respects uniform distance and uniform continuity moduli.

Completely regular topometric spaces and Stone-Čech compactification

Completions

An abstract characterisation of continuous 1-Lipschitz functions