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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Nearness structures and proximity extensions


Authors: M. S. Gagrat and W. J. Thron
Journal: Trans. Amer. Math. Soc. 208 (1975), 103-125
MSC: Primary 54E05
MathSciNet review: 0385808
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Abstract: Proximity, contiguity and nearness structures are here studied from a unified point of view. In the discussion the role that grills can play in the theory is emphasized. Nearness structures were recently introduced by Herrlich and Naimpally. Thron pointed out the importance of grills in proximity theory. Nearness structures $ v$ are then used to generate proximity extensions $ (\phi ,({X^v},{\Pi ^v}))$ of a given LO-proximity space $ (X,\Pi )$, where $ {\Pi _v} = \Pi $. Finally, the relation of the extensions $ (\phi ,({X^v},{\Pi ^v}))$ to arbitrary extensions $ (i,(Y,{\Pi ^ \ast }))$ is investigated.


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DOI: https://doi.org/10.1090/S0002-9947-1975-0385808-3
Keywords: Proximity structures, contiguity structures, nearness structures, proximity mappings, proximity extensions of proximity spaces, strict extensions, principal extensions, trace system, dual trace system, grill, $ \lambda $-clan, $ \lambda $-bunch, $ \lambda $-cluster
Article copyright: © Copyright 1975 American Mathematical Society