© 1995 by London Mathematical Society
© The London Mathematical Society
Optimal Cardinals for Metrizable Barrelled Spaces
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
Received 2 March 1993. Revision received 29 June 1993.
We seek the smallest or largest cardinals for which certain basic results hold, as did Mazur when he proved that c is the smallest infinite-dimensionality for a Fréchet space. As with Mazur, we make no axiomatic assumptions outside the usual ZFC model. We discover three instances in which the optimal cardinal is the dominating number 
and three in which it is the bounding number b, apparently giving the first locally convex space characterizations of these venerable and easily described cardinals. Here are two samples: it is known that for any non-normable metrizable locally convex space E, the minimal size b(E) for a 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-normable metrizable barrelled space of minimal dimension, then 1 dim (E) c; we prove that dim(E) = b. The most important individual result is the reconstruction of Tweddle's space 
without use of the Continuum Hypothesis (1 = c). The reconstruction is vital in the characterizations of b and in subsequent papers answering open questions about countable enlargements.
This paper was written while the second author stayed at the University of Florida supported by DGICYT, BE91-332. He is grateful to UF for its hospitality.