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Proceedings of the American Mathematical Society

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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.

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

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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

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