Nearnesses, proximities, and 𝑇₁-compactifications

Reed, Ellen E.

doi:10.1090/S0002-9947-1978-0474214-1

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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