Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On functions that approximate relations


Author: Gerald Beer
Journal: Proc. Amer. Math. Soc. 88 (1983), 643-647
MSC: Primary 54C60; Secondary 41A65, 54B20
MathSciNet review: 702292
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ X$ be a metric space and let $ Y$ be a separable metric space. Suppose $ R$ is a relation in $ X \times Y$. The following are equivalent: (a) for each $ \varepsilon > 0$ there exists $ f:X \to Y$ such that the Hausdorff distance from $ f$ to $ R$ is at most $ \varepsilon $; (b) the domain of $ R$ is a dense subset of $ X$, and for each isolated point $ x$ of the domain the vertical section of $ R$ at $ x$ is a singleton; (c) for each $ \varepsilon > 0$ there exists $ f:X \to Y$ of Baire class one such that the Hausdorff distance from $ f$ to $ R$ is at most $ \varepsilon $.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54C60, 41A65, 54B20

Retrieve articles in all journals with MSC: 54C60, 41A65, 54B20


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1983-0702292-8
PII: S 0002-9939(1983)0702292-8
Keywords: Approximate selection, Hausdorff metric, functions of Baire class one
Article copyright: © Copyright 1983 American Mathematical Society