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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Nearnesses, proximities, and $T_{1}$-compactifications
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by Ellen E. Reed PDF
Trans. Amer. Math. Soc. 236 (1978), 193-207 Request permission

Abstract:

Gagrat, Naimpally, and Thron together have shown that separated Lodato proximities yield ${T_1}$-compactifications, and conversely. This correspondence is not $1 - 1$, since nonequivalent compactifications can induce the same proximity. Herrlich has shown that if the concept of proximity is replaced by that of nearness then all principal (or strict) ${T_1}$-extensions can be accounted for. (In general there are many nearnesses compatible with a given proximity.) In this paper we obtain a 1-1 correspondence between principal ${T_1}$-extensions and cluster-generated nearnesses. This specializes to a 1-1 match between principal ${T_1}$-compactifications and contigual nearnesses. These results are utilized to obtain a 1-1 correspondence between Lodato proximities and a subclass of ${T_1}$-compactifications. Each proximity has a largest compatible nearness, which is contigual. The extension induced by this nearness is the construction of Gagrat and Naimpally and is characterized by the property that the dual of each clan converges. Hence we obtain a 1-1 match between Lodato proximities and clan-complete principal ${T_1}$-compactifications. When restricted to EF-proximities, this correspondence yields the usual map between ${T_2}$-compactifications and EF-proximities.
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 236 (1978), 193-207
  • MSC: Primary 54E05
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0474214-1
  • MathSciNet review: 0474214