Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-1983-0702292-8
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?)

  • [1] J. P. Aubin, Applied abstract analysis, Wiley, New York, 1977. MR 470034 (81e:54001)
  • [2] G. Beer, On a theorem of Deutsch and Kenderov (submitted).
  • [3] A. Cellina, A further result on the approximation of set valued mappings, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 48 (1970), 412-416. MR 0276935 (43:2675)
  • [4] F. Deutsch and P. Kenderov, Continuous selections and approximate selections for set-valued mappings and applications to metric projections, SIAM J. Math. Anal. 14 (1983), 185-194. MR 686245 (84c:54026)
  • [5] J. Dugundji, Topology, Allyn and Bacon, Boston, Mass., 1966. MR 0193606 (33:1824)
  • [6] K. Kuratowski, Topology, Vol. 1, Academic Press, New York, 1966. MR 0217751 (36:840)
  • [7] E. Michael, Selected selection theorems, Amer. Math. Monthly 63 (1956), 233-237. MR 1529282
  • [8] D. Montgomery, Nonseparable metric spaces, Fund. Math. 25 (1935), 527-533.
  • [9] S. Nadler, Hyperspaces of sets, Dekker, New York, 1978. MR 0500811 (58:18330)

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: 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

American Mathematical Society