Optimal Cardinals for Metrizable Barrelled Spaces |
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Authors: | Saxon SA; Sanchez Ruiz LM |
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Institution: | Department of Mathematics, University of Florida PO Box 118000, Gainesville, Florida 32611-8000, USA E-mail: saxon{at}math.ufl.edu
EUITI-Departamento de Matemática Aplicada, Universidad Politécnica de Valencia 46071 Valencia, Spain |
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Abstract: | We seek the smallest or largest cardinals for which certainbasic results hold, as did Mazur when he proved that c is thesmallest infinite-dimensionality for a Fréchet space.As with Mazur, we make no axiomatic assumptions outside theusual ZFC model. We discover three instances in which the optimalcardinal is the dominating number and three in which it isthe bounding number b, apparently giving the first locally convexspace characterizations of these venerable and easily describedcardinals. Here are two samples: it is known that for any non-normablemetrizable locally convex space E, the minimal size b(E) fora fundamental system of bounded sets must satisfy 1 b(E) c;we prove that b(E) = . Again, it is known that if E is a non-normablemetrizable barrelled space of minimal dimension, then 1 dim(E) c; we prove that dim(E) = b. The most important individualresult is the reconstruction of Tweddle's space without useof the Continuum Hypothesis (1 = c). The reconstruction is vitalin the characterizations of b and in subsequent papers answeringopen questions about countable enlargements. |
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