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A Theory of Convergence

Published online by Cambridge University Press:  20 November 2018

E. J. McShane
E. J. McShane*
Affiliation:
University of Virginia
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The literature already contains several theories of limits which have great generality (10; 12; 1; 5; 6; 13; 3; 14; 4, p. 34). Nevertheless, the intrinsic importance and frequent use of the concept may justify the publication of another variant, provided that it has advantages in ease of application without sacrifice of generality. The theory of convergence studied in this note includes the other theories and their applications in a smooth way, without artifice.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 6 , 1954 , pp. 161 - 168
Copyright
Copyright © Canadian Mathematical Society 1954

References

1. Bennett, A. A., Generalized convergence with binary relations, Amer. Math. Monthly, 32 (1925), 131–134.Google Scholar
2. Birkhoff, G., Moore-Smith convergence in general topology, Ann. Math. (2), 38 (1937), 39–60.Google Scholar
3. Birkhoff, G., An ergodic theorem for general semi-groups, Proc. Nat. Acad. Sci., 25 (1939), 625–627.Google Scholar
4. Bourbaki, N., Éléments de mathématique, II, Livre III, chap. I (Paris, Hermann et Cie., Actualités Scientifiques et Industrielles No. 858, 1940).Google Scholar
5. Cartan, H., Théorie des filtres, Comptes Rendus de l'Ac. des Sci. (Paris), 205 (1937), 595–598.Google Scholar
6. Cartan, H., Filtres et ultrafiltres, Comptes Rendus de l'Ac. des Sci. (Paris), 205 (1937), 777–779.Google Scholar
7. Kelley, J. L., Convergence in topology, Duke Math. Jour., 17 (1950), 277–283.Google Scholar
8. McShane, E. J., Partial orderings and Moore-Smith limits, Amer. Math. Monthly, 59 (1952), 1–11.Google Scholar
9. McShane, E. J., Order-preserving maps and integration (Princeton Univ. Press, Annals of Math. Studies No. 31, 1953).Google Scholar
10. Moore, E. H., Definition of limit in general integral analysis, Proc. Nat. Acad. Sci., 1 (1915), 628–632.Google Scholar
11. Moore, E. H., General analysis, vol. I, pt. n, with the cooperation of R. W. Barnard (American Philosophical Society, 1939).Google Scholar
12. Moore, E. H. and Smith, H. L., A general theory of limits, Amer. J. Math. 44 (1922), 102–121.Google Scholar
13. Smith, H. L., A general theory of limits, National Math. Magazine, 12 (1937-38), 371–379.Google Scholar
14. Tukey, J. W., Convergence and uniformity in general topology (Princeton, 1940).Google Scholar