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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
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Abstract:

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.


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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