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

Open and uniformly open relations


Authors: P. Mah and S. A. Naimpally
Journal: Proc. Amer. Math. Soc. 66 (1977), 159-166
MSC: Primary 54E05
MathSciNet review: 0454925
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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.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1977-0454925-9
PII: S 0002-9939(1977)0454925-9
Keywords: Open relations, nearly open, weakly open, Morita uniformity, proximity, uniformly open, uniformly nearly open, nearly continuous, closed graph
Article copyright: © Copyright 1977 American Mathematical Society