Open and uniformly open relations

Authors:
P. Mah and S. A. Naimpally

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

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Abstract | References | Similar Articles | Additional Information

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.

**[1]**John L. Kelley,*General topology*, D. Van Nostrand Company, Inc., Toronto-New York-London, 1955. MR**0070144****[2]**Kiiti Morita,*On the simple extension of a space with respect to a uniformity. I*, Proc. Japan Acad.**27**(1951), 65–72. MR**0048782****[3]**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****[4]**B. J. Pettis,*Closed graph and open mapping theorems in certain topologically complete spaces*, Bull. London Math. Soc.**6**(1974), 37–41. MR**0450929**, https://doi.org/10.1112/blms/6.1.37**[5]**-,*Some topological questions related to open and closed graph theorems*, Studies in Topology, Academic Press, New York, 1975, pp. 451-456.**[6]**V. Z. Poljakov,*Open mappings of proximity spaces*, Dokl. Akad. Nauk SSSR**155**(1964), 1014–1017 (Russian). MR**0172240****[7]**J. D. Weston,*On the comparison of topologies*, J. London Math. Soc.**32**(1957), 342–354. MR**0094776**, https://doi.org/10.1112/jlms/s1-32.3.342**[8]**Colloquium Co-topology 1964-1965, Mathematical Centre (Amsterdam) [Notes by J. M. Aarts].

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0454925-9

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