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

DOI:
https://doi.org/10.1090/S0002-9939-00-05790-7

Published electronically:
December 4, 2000

MathSciNet review:
1825920

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Let be an infinite set, a set of pseudo-metrics on and If is limited (finite) for every and every then, for each we can define a pseudo-metric on by writing st We investigate the conditions under which the topology induced on by 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.

**1.**Abraham Robinson,*Non-standard analysis*, North-Holland Publishing Co., Amsterdam, 1966. MR**0205854****2.**C. Ward Henson,*The nonstandard hulls of a uniform space*, Pacific J. Math.**43**(1972), 115–137. MR**0314007****3.**K. D. Stroyan and W. A. J. Luxemburg,*Introduction to the theory of infinitesimals*, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. Pure and Applied Mathematics, No. 72. MR**0491163****4.**W. A. J. Luxemburg,*A general theory of monads*, Applications of Model Theory to Algebra, Analysis, and Probability (Inte rnat. Sympos., Pasadena, Calif., 1967) Holt, Rinehart and Winston, New York, 1969, pp. 18–86. MR**0244931****5.**Stephen Willard,*General topology*, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1970. MR**0264581****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 (Szekszárd, 1993) Bolyai Soc. Math. Stud., vol. 4, János Bolyai Math. Soc., Budapest, 1995, pp. 413–418. MR**1374820****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**

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Additional Information

**Nader Vakil**

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

Email:
N-Vakil@wiu.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05790-7

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