Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Nonstandard topologies with bases that consist only of standard sets

Author: Nader Vakil
Journal: Proc. Amer. Math. Soc. 129 (2001), 2075-2083
MSC (2000): Primary 46S20; Secondary 54J05
Published electronically: December 4, 2000
MathSciNet review: 1825920
Full-text PDF

Abstract | References | Similar Articles | Additional Information


Let $X$ be an infinite set, $D$ a set of pseudo-metrics on $X,$ $\Xi\subset ^*X,$ and $\Gamma\subset ^*D.$ If $\rho (a,b)$ is limited (finite) for every $a,b\in \Xi$ and every $\rho\in \Gamma,$ then, for each $\rho\in \Gamma,$ we can define a pseudo-metric $\tilde\rho$ on $\Xi$ by writing $\tilde\rho(a,b)=\,$st $(\rho(a,b)).$ We investigate the conditions under which the topology induced on $\Xi$ by $\{\tilde\rho: \rho\in \Gamma\}$ has a basis consisting only of standard sets. This investigation produces a theory with a variety of applications in functional analysis. For example, a specialization of some of our general results will yield such classical compactness theorems as Schauder's theorem, Mazur's theorem, and Gelfand-Philips's theorem.

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

  • 1. Abraham Robinson, Non-Standard Analysis, North-Holland Publishing Company, Amsterdam, 1966. MR 34:5680
  • 2. C. Ward Henson, The Nonstandard Hulls of a Uniform Space, Pacific Journal of Mathematics, vol. 43, No. 1, 1972, pp. 115-137. MR 47:2559
  • 3. W. A. J. Luxemburg and K. D. Stroyan,, Introduction to the theory of infinitesimals, Academic Press, New York, 1976. MR 58:10429
  • 4. W. A. J. Luxemburg, A general theory of monads, Applications of model theory to algebra, analysis, and probability (W. A. J. Luxemburg, editor), Holt, Reinhart and Winston, New York, 1969. MR 39:6244
  • 5. Stephen Willard, General Topology, Addison-Wesley Publishing Company, Reading, Massachusetts, 1970. MR 41:9173
  • 6. S. Salbany and T. D. Todorov, Nonstandard Analysis in Topology: Nonstandard and Standard Compactifications, to appear in J. Symb. Logic.
  • 7. H. Render, Topologies on The Nonstandard Model, Topology with Applications, Bolyai Society, Mathematical Studies, 4, Szekszard(Hungry), 1993. pp. 413-418. MR 97a:54052
  • 8. R. G. Bartle, On Compactness in Functional Analysis, Transactions of the American Mathematical Society, vol. 79, 1955, pp. 35-57. MR 17:510h
  • 9. S. Kakutani, A proof of Schauder's theorem, Journal of the Mathematical Society of Japan, vol. 3, No. 1, May 1951, pp. 228-231. MR 13:355e

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46S20, 54J05

Retrieve articles in all journals with MSC (2000): 46S20, 54J05

Additional Information

Nader Vakil
Affiliation: Department of Mathematics, Western Illinois University, Macomb, Illinois 61455

Keywords: Nonstandard topologies, Banach space, compactness conditions
Received by editor(s): June 8, 1999
Received by editor(s) in revised form: November 16, 1999
Published electronically: December 4, 2000
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society