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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

$ U$-embedded subsets of normed linear spaces


Authors: Ronnie Levy and M. D. Rice
Journal: Proc. Amer. Math. Soc. 97 (1986), 727-733
MSC: Primary 54E15; Secondary 54C30, 54C45
MathSciNet review: 845997
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Abstract: A subset $ S$ of a metric space $ X$ is $ U$-embedded in $ X$ if every uniformly continuous function $ f:S \to R$ extends to a uniformly continuous function $ F:X \to R$. Thus $ U$-embedding is the uniform analogue of $ C$-embedding. The Tietze extension theorem tells us exactly which subsets of metric spaces are $ C$-embedded. The uniform analogue would tell us exactly which subsets of metric spaces are $ U$-embedded. In this paper, a characterization of $ U$-embedded subsets of the Euclidean plane (or any normed linear space) is given.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0845997-3
PII: S 0002-9939(1986)0845997-3
Keywords: $ U$-embedding, Lipschitz for large distances
Article copyright: © Copyright 1986 American Mathematical Society