Definable Functions in Urysohn's Metric Space

Isaac Goldbring

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arXiv · 1001.4999

Definable Functions in Urysohn's Metric Space

Isaac Goldbring

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Abstract

Let U denote the Urysohn sphere and consider U as a metric structure in the empty continuous signature. We prove that every definable function from U^n to 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.

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Isaac Goldbring. 2010-01-27. Definable Functions in Urysohn's Metric Space. https://arxiv.org/abs/1001.4999

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