Nearness structures and proximity extensions

Gagrat, M. S.; Thron, W. J.

doi:10.1090/S0002-9947-1975-0385808-3

Nearness structures and proximity extensions
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by M. S. Gagrat and W. J. Thron

Trans. Amer. Math. Soc. 208 (1975), 103-125

DOI: https://doi.org/10.1090/S0002-9947-1975-0385808-3

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Abstract:

Proximity, contiguity and nearness structures are here studied from a unified point of view. In the discussion the role that grills can play in the theory is emphasized. Nearness structures were recently introduced by Herrlich and Naimpally. Thron pointed out the importance of grills in proximity theory. Nearness structures $v$ are then used to generate proximity extensions $(\phi ,({X^v},{\Pi ^v}))$ of a given LO-proximity space $(X,\Pi )$, where ${\Pi _v} = \Pi$. Finally, the relation of the extensions $(\phi ,({X^v},{\Pi ^v}))$ to arbitrary extensions $(i,(Y,{\Pi ^ \ast }))$ is investigated.

References

Bernhard Banaschewski, Extensions of topological spaces, Canad. Math. Bull. 7 (1964), 1–22. MR 161293, DOI 10.4153/CMB-1964-001-5

Extensions of maps on nearness spaces

Nearness spaces and extensions of topological spaces

Eduard Čech, Topological spaces, Publishing House of the Czechoslovak Academy of Sciences, Prague; Interscience Publishers John Wiley & Sons, London-New York-Sydney, 1966. Revised edition by Zdeněk Frolík and Miroslav Katětov; Scientific editor, Vlastimil Pták; Editor of the English translation, Charles O. Junge. MR 0211373
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
Horst Herrlich, A concept of nearness, General Topology and Appl. 4 (1974), 191–212. MR 350701, DOI 10.1016/0016-660X(74)90021-X

Topological structures

V. M. Ivanova and A. A. Ivanov, Contiguity spaces and bicompact extensions of topological spaces, Izv. Akad. Nauk SSSR Ser. Mat. 23 (1959), 613–634 (Russian). MR 0110082

Regular extensions of topological spaces

S. Leader, On clusters in proximity spaces, Fund. Math. 47 (1959), 205–213. MR 112120, DOI 10.4064/fm-47-2-205-213
Michael W. Lodato, On topologically induced generalized proximity relations. II, Pacific J. Math. 17 (1966), 131–135. MR 192470, DOI 10.2140/pjm.1966.17.131
S. A. Naimpally, Reflective functors via nearness, Fund. Math. 85 (1974), 245–255. MR 353244, DOI 10.4064/fm-85-3-245-255
S. A. Naimpally and J. H. M. Whitfield, Not every near family is contained in a clan, Proc. Amer. Math. Soc. 47 (1975), 237–238. MR 358703, DOI 10.1090/S0002-9939-1975-0358703-9

Proximity extensions

A proximity without a smallest compatible nearness

Yu. M. Smirnov, On proximity spaces, Mat. Sbornik N.S. 31(73) (1952), 543–574 (Russian). MR 0055661

On contiguity spaces

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
F. J. Wagner, Note on compactification. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 60 = Indag. Math. 19 (1957), 171–176, 177–181. MR 0087919, DOI 10.1016/S1385-7258(57)50022-X