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Erschienen in:
2013 | OriginalPaper | Buchkapitel
Oscillation of Urysohn Type Spaces
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Abstract
A metric space \(\mathrm{M} = (M;d )\) is 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 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 oscillation stable if for every \(\epsilon > 0\) and every uniformly continuous and bounded function \(f : M \rightarrow \mathfrak{R}\) there exists an isometric copy \(\mathrm{{M}}^{{\ast}} = ({M}^{{\ast}};d )\) of \(\mathrm{M}\) in \(\mathrm{M}\) for which:
$$\sup \{\vert f(x) - f(y)\vert \mid x,y \in {M}^{{\ast}}\} < \epsilon.$$
Theorem.
Every bounded, uncountable, separable, complete, homogeneous, universal metric space
\(\mathrm{M} = (M;d )\)
is oscillation stable. (Theorem 12.)