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: 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]**J. L. Kelley,*General topology*, Van Nostrand, Princeton, N. J., 1955. MR**0070144 (16:1136c)****[2]**K. Morita,*On the simple extension of a space with respect to a uniformity*. I, Proc. Japan Acad.**27**(1951), 65-72. MR**0048782 (14:68f)****[3]**S. A. Naimpally and B. D. Warrack,*Proximity spaces*, Cambridge Univ. Press, Cambridge, 1970. MR**0278261 (43:3992)****[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 (56:9219)****[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*, Soviet Math. Dokl.**5**(1964), 545-548. MR**0172240 (30:2460)****[7]**J. D. Weston,*On the comparison of topologies*, J. London Math. Soc.**32**(1957), 342-354. MR**0094776 (20:1288)****[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