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On functions that approximate relations


Author: Gerald Beer
Journal: Proc. Amer. Math. Soc. 88 (1983), 643-647
MSC: Primary 54C60; Secondary 41A65, 54B20
DOI: https://doi.org/10.1090/S0002-9939-1983-0702292-8
MathSciNet review: 702292
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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 $.


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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0702292-8
Keywords: Approximate selection, Hausdorff metric, functions of Baire class one
Article copyright: © Copyright 1983 American Mathematical Society