Illinois Journal of Mathematics

Definable functions in Urysohn’s metric space

Isaac Goldbring

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Let $\mathfrak{U}$ denote the Urysohn sphere and consider $\mathfrak{U}$ as a metric structure in the empty continuous signature. We prove that every definable function $\mathfrak{U}^{n}\to\mathfrak{U}$ is either a projection function or else has relatively compact range. As a consequence, we prove that many functions natural to the study of the Urysohn sphere are not definable. We end with further topological information on the range of the definable function in case it is compact.

Article information

Illinois J. Math., Volume 55, Number 4 (2011), 1423-1435.

First available in Project Euclid: 12 July 2013

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Zentralblatt MATH identifier

Primary: 03C40: Interpolation, preservation, definability


Goldbring, Isaac. Definable functions in Urysohn’s metric space. Illinois J. Math. 55 (2011), no. 4, 1423--1435. doi:10.1215/ijm/1373636691.

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