Illinois Journal of Mathematics

Definable functions in Urysohn’s metric space

Isaac Goldbring

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Abstract

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

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

Dates
First available in Project Euclid: 12 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1373636691

Digital Object Identifier
doi:10.1215/ijm/1373636691

Mathematical Reviews number (MathSciNet)
MR3082876

Zentralblatt MATH identifier
1279.03061

Subjects
Primary: 03C40: Interpolation, preservation, definability

Citation

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


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