Oscillation of Urysohn type spaces

TL;DR

This paper investigates the oscillation stability of certain universal, homogeneous metric spaces, establishing that all such uncountable, complete, separable spaces exhibit this stability property.

Contribution

It proves that every uncountable, complete, separable, homogeneous, universal metric space is oscillation stable, extending understanding of stability in Urysohn-type spaces.

Findings

Uncountable, complete, separable, homogeneous, universal metric spaces are oscillation stable.

Oscillation stability holds for all such spaces, regardless of specific structure.

The result generalizes stability properties in metric space theory.

Abstract

A metric space $\mathrm{M}=(M;\de)$ is {\em homogeneous} if for every isometry $\alpha$ of a finite subspace of $\mathrm{M}$ to a subspace of $\mathrm{M}$ there exists an isometry of $\mathrm{M}$ onto $\mathrm{M}$ extending $\alpha$. The metric space $\mathrm{M}$ is {\em universal} if it isometrically embeds every finite metric space $\mathrm{F}$ with $\dist(\mathrm{F})\subseteq \dist(\mathrm{M})$. ($\dist(\mathrm{M})$ being the set of distances between points of $\mathrm{M}$.) A metric space $\mathrm{M}$ is {\em oscillation stable} if for every $\epsilon>0$ and every uniformly continuous and bounded function $f: M\to \Re$ there exists an isometric copy $\mathrm{M}^\ast=(M^\ast; \de)$ of $\mathrm{M}$ in $\mathrm{M}$ for which: \[ \sup\{|f(x)-f(y)| \mid x,y\in M^\ast\}<\epsilon. \] Every bounded, uncountable, separable, complete, homogeneous, universal metric space $\mathrm{M}=(M;\de)$ is oscillation stable. (Theorem thm:finabstr.)