Open and uniformly open relations
Authors:
P. Mah and S. A. Naimpally
Journal:
Proc. Amer. Math. Soc. 66 (1977), 159166
MSC:
Primary 54E05
MathSciNet review:
0454925
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Abstract: It is shown that if is an Efremovič proximity space, Y is a topological space, is an injective relation, then R is open if and only if R is weakly open, nearly open and 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 . These results include, as special cases, results of Kelley, Pettis and Weston.
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 B. J. Pettis, Closed graph and open mapping theorems in certain topologically complete spaces, Bull. London Math. Soc. 6 (1974), 3741. MR 0450929 (56:9219)
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 , Some topological questions related to open and closed graph theorems, Studies in Topology, Academic Press, New York, 1975, pp. 451456.
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 V. Z. Poljakov, Open mappings of proximity spaces, Soviet Math. Dokl. 5 (1964), 545548. MR 0172240 (30:2460)
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 J. D. Weston, On the comparison of topologies, J. London Math. Soc. 32 (1957), 342354. MR 0094776 (20:1288)
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 Colloquium Cotopology 19641965, Mathematical Centre (Amsterdam) [Notes by J. M. Aarts].
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939197704549259
PII:
S 00029939(1977)04549259
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
