Open and uniformly open relations

Mah, P.; Naimpally, S. A.

doi:10.1090/S0002-9939-1977-0454925-9

Open and uniformly open relations
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by P. Mah and S. A. Naimpally

Proc. Amer. Math. Soc. 66 (1977), 159-166

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

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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

John L. Kelley, General topology, D. Van Nostrand Co., Inc., Toronto-New York-London, 1955. MR 0070144
Kiiti Morita, On the simple extension of a space with respect to a uniformity. I, Proc. Japan Acad. 27 (1951), 65–72. MR 48782
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
B. J. Pettis, Closed graph and open mapping theorems in certain topologically complete spaces, Bull. London Math. Soc. 6 (1974), 37–41. MR 450929, DOI 10.1112/blms/6.1.37

Some topological questions related to open and closed graph theorems

V. Z. Poljakov, Open mappings of proximity spaces, Dokl. Akad. Nauk SSSR 155 (1964), 1014–1017 (Russian). MR 0172240
J. D. Weston, On the comparison of topologies, J. London Math. Soc. 32 (1957), 342–354. MR 94776, DOI 10.1112/jlms/s1-32.3.342