Proximity approach to extension problems

Gagrat, M. S.; Naimpally, S. A.

doi:10.4064/fm-71-1-63-76

Cited by 38 publications

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“…Perhaps his greatest achievement here with M.S. Gagrat was his description of a general method to obtain all T 1 compactifications of a T 1 space determined by different Lodato proximities on the space, the points of the space being all maximal bunches equipped with the Wallman topology [119]. A bunch in a Lodato proximity space X, δ is a nonempty family of nonempty subsets σ satisfying these three properties:…”

Section: Some Research Highlightsmentioning

confidence: 99%

Somashekhar Naimpally, 1931–2014

Beer

Concilio

Maio³

et al. 2015

Topology and its Applications

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Section: Some Research Highlightsmentioning

confidence: 99%

Somashekhar Naimpally, 1931–2014

Beer

Concilio

Maio³

et al. 2015

Topology and its Applications

Self Cite

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“…If (X, ) and (W, ) are nearness spaces and if u : X → W is a map, then, in [1], Bentley defined a map u + from the family of -bunches to the family of -bunches. An analogue of u + in an LO-proximity space is given by Gagrat and Naimpally [3]. However, -bunches, if defined, does not work smoothly in an L-merotopic space (L X , ), due to the nonavailability of the condition:…”

Section: Introductionmentioning

confidence: 99%

Complete -grills and -merotopies

Khare

Singh

2008

Fuzzy Sets and Systems

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“…The next theorem verifies that this is the construction of Gagrat and Naimpally [3,Theorem 3.13]. For this reason we define a GN-compactification to be any compactification which is equivalent to some Ext(vc(w)), where it is a compatible Lodato proximity.…”

Section: Proximities and Extensions Each Proximity It Carries Wmentioning

confidence: 99%

“…Naimpally [3]. These compactifications are characterized by the property that the dual of each clan converges.…”

Section: Proximities and Extensions Each Proximity It Carries Wmentioning

confidence: 99%

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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Proximity approach to extension problems

Cited by 38 publications

References 0 publications

Somashekhar Naimpally, 1931–2014

Somashekhar Naimpally, 1931–2014

Complete -grills and -merotopies

Nearnesses, proximities, and 𝑇₁-compactifications

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