Nearnesses, proximities, and $T_{1}$-compactifications
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- by Ellen E. Reed
- Trans. Amer. Math. Soc. 236 (1978), 193-207
- DOI: https://doi.org/10.1090/S0002-9947-1978-0474214-1
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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.References
- Bernhard Banaschewski, Extensions of topological spaces, Canad. Math. Bull. 7 (1964), 1–22. MR 161293, DOI 10.4153/CMB-1964-001-5
- H. L. Bentley, Nearness spaces and extensions of toplogical spaces, Studies in topology (Proc. Conf., Univ. North Carolina, Charlotte, N.C., 1974; dedicated to Math. Sect. Polish Acad. Sci.), Academic Press, New York, 1975, pp. 47–66. MR 0380738
- M. S. Gagrat and S. A. Naimpally, Proximity approach to extension problems, Fund. Math. 71 (1971), no. 1, 63–76. (errata insert). MR 293576, DOI 10.4064/fm-71-1-63-76
- M. S. Gagrat and W. J. Thron, Nearness structures and proximity extensions, Trans. Amer. Math. Soc. 208 (1975), 103–125. MR 385808, DOI 10.1090/S0002-9947-1975-0385808-3
- Horst Herrlich, A concept of nearness, General Topology and Appl. 4 (1974), 191–212. MR 350701, DOI 10.1016/0016-660X(74)90021-X
- Michael W. Lodato, On topologically induced generalized proximity relations, Proc. Amer. Math. Soc. 15 (1964), 417–422. MR 161305, DOI 10.1090/S0002-9939-1964-0161305-3
- Yu. M. Smirnov, On proximity spaces, Mat. Sbornik N.S. 31(73) (1952), 543–574 (Russian). MR 0055661
- Wolfgang J. Thron, Topological structures, Holt, Rinehart and Winston, New York-Toronto-London, 1966. MR 0200892
- W. J. Thron, Proximity structures and grills, Math. Ann. 206 (1973), 35–62. MR 336710, DOI 10.1007/BF01431527
- W. J. Thron, On a problem of F. Riesz concerning proximity structures, Proc. Amer. Math. Soc. 40 (1973), 323–326. MR 322818, DOI 10.1090/S0002-9939-1973-0322818-X
Bibliographic 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