Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Nearness structures and proximity extensions


Authors: M. S. Gagrat and W. J. Thron
Journal: Trans. Amer. Math. Soc. 208 (1975), 103-125
MSC: Primary 54E05
DOI: https://doi.org/10.1090/S0002-9947-1975-0385808-3
MathSciNet review: 0385808
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] B. Banaschewski, Extensions of topological spaces, Canad. Math. Bull. 7 (1964), 1-22. MR 28 #4501. MR 0161293 (28:4501)
  • [2] H. L. Bentley, Extensions of maps on nearness spaces (preprint).
  • [3] -, Nearness spaces and extensions of topological spaces (preprint).
  • [4] E. Čech, Topological spaces, rev. ed., Publ. House Czech. Acad. Sci., Prague; English transl., Wiley, New York, 1966. MR 21 #2962; 35 #2254. MR 0211373 (35:2254)
  • [5] M. S. Gagrat and S. A. Naimpally, Proximity approach to extension problems, Fund. Math. 71 (1971), 63-76. MR #2653. MR 0293576 (45:2653)
  • [6] H. Herrlich, A concept of nearness, General Topology and its Appl. (to appear). MR 0350701 (50:3193)
  • [7] -, Topological structures (preprint).
  • [8] 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 22 #965. MR 0110082 (22:965)
  • [9] A. A. Ivanov, Regular extensions of topological spaces, Contributions to Extension Theory of Topological Structures, Berlin, 1969, pp. 133-138.
  • [10] S. Leader, On clusters in proximity spaces, Fund. Math. 47 (1959), 205-213. MR 22 #2978. MR 0112120 (22:2978)
  • [11] M. W. Lodato, On topologically induced generalized proximity relations. II, Pacific J. Math. 17 (1966), 131-135. MR 33 #695. MR 0192470 (33:695)
  • [12] S. A. Naimpally, Reflective functors via nearness, Fund. Math. (to appear). MR 0353244 (50:5728)
  • [13] 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 0358703 (50:11162)
  • [14] -, Proximity extensions (preprint).
  • [15] O. Njåstad, A proximity without a smallest compatible nearness (preprint).
  • [16] Ju. M. Smirnov, On proximity spaces, Mat. Sb. 31 (73) (1952), 543-574; English transl., Amer. Math. Soc. Transl. (2) 38 (1964), 5-35. MR 14, 1107. MR 0055661 (14:1107b)
  • [17] W. L. Terwilliger, On contiguity spaces, Ph.D. Dissertation, Washington State University, Pullman, Washington, 1965.
  • [18] W. J. Thron, Topological structures, Holt, Rinehart and Winston, New York, 1966. MR 34 #778. MR 0200892 (34:778)
  • [19] -, Proximity structures and grills, Math. Ann. 206 (1973), 35-62. MR 0336710 (49:1483)
  • [20] -, On a problem of F. Riesz concerning proximity structures, Proc. Amer. Math. Soc. 40 (1973), 323-326. MR 48 #1179. MR 0322818 (48:1179)
  • [21] F. J. Wagner, Notes on compactification. I, II, Nederl. Akad. Wetensch. Proc. Ser. A 60 = Indag. Math. 19 (1957), 171-181. MR 19, 436. MR 0087919 (19:436c)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 54E05

Retrieve articles in all journals with MSC: 54E05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1975-0385808-3
Keywords: Proximity structures, contiguity structures, nearness structures, proximity mappings, proximity extensions of proximity spaces, strict extensions, principal extensions, trace system, dual trace system, grill, $ \lambda $-clan, $ \lambda $-bunch, $ \lambda $-cluster
Article copyright: © Copyright 1975 American Mathematical Society

American Mathematical Society