Oscillation of Urysohn type spaces

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.)