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Nearnesses and T0-Extensions of Topological Spaces

Published online by Cambridge University Press:  20 November 2018

Alice M. Dean
Alice M. Dean*
Affiliation:
Department of Mathematics, Bates College, LewistonMaine 04240
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Abstract

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In [3], Reed establishes a bijection between the (equivalence classes of) principal T1-extensions of a topological space X and the compatible, cluster-generated, Lodato nearnesses on X. We extend Reed's result to the T0 case by obtaining a one-to-one correspondence between the principal T0-extensions of a space X and the collections of sets (called “t-grill sets”) which generate a certain class of nearnesses which we call “t-bunch generated” nearnesses. This correspondence specializes to principal T0-compactifications. Finally, we show that there is a bijection between these t-grill sets and the filter systems of Thron [5], and that the corresponding extensions are equivalent.

Keywords

54E1754A0554C2054D1054D35PrincipalT0 -extensionnearnesst -Lodatogrillt-bunch generatedt-grill setcompactificationcontigualcompatibleLodatocluster
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 26 , Issue 4 , 01 December 1983 , pp. 430 - 437
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Bentley, H. L. and Herrlich, H., Extensions of Topological Spaces, Topology, Lecture Notes in Pure and Appl. Math. 24, Dekker, New York, 1976, 129-184.Google Scholar
2. Herrlich, H., A Concept of Nearness, Gen. Top. Appl. 4 191-212 (1974).Google Scholar
3. Reed, E. E., Nearnesses, proximities, and T1compactifications, Trans. Amer. Math. Soc. 236 193-207 (1978).Google Scholar
4. Thron, W. J., Proximity Structures and Grills, Math. Ann. 206 35-62 (1973).Google Scholar
5. Thron, W. J., Topological Structures, Holt, Rinehart, and Winston, New York, 1966.Google Scholar