Nearness Spaces and Extensions of Topological Spaces

Bentley, H. L.

doi:10.1016/b978-0-12-663450-1.50012-8

Cited by 38 publications

(51 citation statements)

References 7 publications

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“…It is similar to a result of Bentley [2] which reaches the same conclusion under a slightly weaker hypothesis [2, Theorems 2.8 and 3.6]. The proof is given here for convenience.…”

supporting

confidence: 86%

Nearnesses, proximities, and 𝑇₁-compactifications

Reed¹

1978

Trans. Amer. Math. Soc.

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Abstract. Gagrat, Naimpally, and Thron together have shown that separated Lodato proximities yield 7",-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) 7",-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 7",-extensions and cluster-generated nearnesses. This specializes to a 1-1 match between principal Trcompactifications and contigua! nearnesses.These results are utilized to obtain a 1-1 correspondence between Lodato proximities and a subclass of T,-compactifications. Each proximity has a largest compatible nearness, which is contigua! 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 r,-compactifications. When restricted to ¿/-proximities, this correspondence yields the usual map between r2-compactifications and ¿/'-proximities.

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“…It is similar to a result of Bentley [2] which reaches the same conclusion under a slightly weaker hypothesis [2, Theorems 2.8 and 3.6]. The proof is given here for convenience.…”

supporting

confidence: 86%

Nearnesses, proximities, and 𝑇₁-compactifications

Reed¹

1978

Trans. Amer. Math. Soc.

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“…(2) follows from Theorem 4 and Theorem A, together with the fact that the completion of a regular nearness space is regular, and its underlying topology is regular [3].…”

Section: Definitionmentioning

confidence: 99%

Developable spaces and nearness structures

Carlson¹

1980

Proc. Amer. Math. Soc.

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Abstract. Necessary and sufficient conditions on a nearness structure are provided for which the underlying topology is developable. It is also shown that a topology is developable if and only if it admits a compatible nearness structure with a countable base. Finally it is shown that a topological space is embeddable in a complete Moore space if and only if it admits a compatible nearness structure satisfying certain stated conditions. Moreover, the complete Moore space is homeomorphic to Herrlich's completion of the space with the specified nearness structure.

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“…Then R {cly(A -(X n ß)): ß E S,} C n {Y -Q: Q E %} = 0. Thus each S, belongs to p. The following important theorem was proved by Bentley and Herrlich in [3]. It is stated here for the convenience of the reader.…”

Section: Definitionmentioning

confidence: 99%

“…Nearness structures were introduced by Herrlich in [5] and have proved useful in unifying various topological structures. Their utility in the study of extensions of a topological space is demonstrated in [2], [3], [4] and [6].…”

Section: Developable Spaces and Nearness Structures John W Carlsonmentioning

confidence: 99%