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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Open and uniformly open relations
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by P. Mah and S. A. Naimpally PDF
Proc. Amer. Math. Soc. 66 (1977), 159-166 Request permission

Abstract:

It is shown that if $(X,\delta )$ is an Efremovič proximity space, Y is a topological space, $R \subset X \times Y$ is an injective relation, then R is open if and only if R is weakly open, nearly open and $R[X]$ is open in Y. An analogous result is proved when X is a uniform space and Y a Morita uniform space: (i) if R is uniformly open, then R is weakly open and uniformly nearly open; (ii) if R is weakly open, and uniformly nearly open, then R is uniformly open on X to $R[X]$. These results include, as special cases, results of Kelley, Pettis and Weston.
References
  • John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
  • Kiiti Morita, On the simple extension of a space with respect to a uniformity. I, Proc. Japan Acad. 27 (1951), 65–72. MR 48782
  • S. A. Naimpally and B. D. Warrack, Proximity spaces, Cambridge Tracts in Mathematics and Mathematical Physics, No. 59, Cambridge University Press, London-New York, 1970. MR 0278261
  • B. J. Pettis, Closed graph and open mapping theorems in certain topologically complete spaces, Bull. London Math. Soc. 6 (1974), 37–41. MR 450929, DOI 10.1112/blms/6.1.37
  • —, Some topological questions related to open and closed graph theorems, Studies in Topology, Academic Press, New York, 1975, pp. 451-456.
  • V. Z. Poljakov, Open mappings of proximity spaces, Dokl. Akad. Nauk SSSR 155 (1964), 1014–1017 (Russian). MR 0172240
  • J. D. Weston, On the comparison of topologies, J. London Math. Soc. 32 (1957), 342–354. MR 94776, DOI 10.1112/jlms/s1-32.3.342
  • Colloquium Co-topology 1964-1965, Mathematical Centre (Amsterdam) [Notes by J. M. Aarts].
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Additional Information
  • © Copyright 1977 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 66 (1977), 159-166
  • MSC: Primary 54E05
  • DOI: https://doi.org/10.1090/S0002-9939-1977-0454925-9
  • MathSciNet review: 0454925