caa a22 4500 397532679 CHVBK 20180308164655.0 cr unu---uuuuu 161202e199502 xx s 000 0 eng 10.1112/jlms/51.1.137 doi (NATIONALLICENCE)oxford-10.1112/jlms/51.1.137 Optimal Cardinals for Metrizable Barrelled Spaces [Elektronische Daten] [S.A. Saxon, L.M. Sánchez Ruiz] 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 𝔡 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 𝔡b(E) for a fundamental system of bounded sets must satisfy ℵ1 ≤ 𝔡b(E) ≤ c; we prove that 𝔡b(E) = 𝔡. Again, it is known that if E is a non-normable metrizable barrelled space of minimal dimension, then ℵ1 ≤ dim (E) ≤ c; we prove that dim(E) = b. The most important individual result is the reconstruction of Tweddle's space ψ without use of the Continuum Hypothesis (ℵ1 = c). The reconstruction is vital in the characterizations of b and in subsequent papers answering open questions about countable enlargements. © The London Mathematical Society Notes and Papers nationallicence Saxon S.A. Department of Mathematics, University of Florida PO Box 118000, Gainesville, Florida 32611-8000, USA E-mail: saxon@math.ufl.edu aut Sánchez Ruiz L.M. EUITI-Departamento de Matemática Aplicada, Universidad Politécnica de Valencia 46071 Valencia, Spain aut Journal of the London Mathematical Society Oxford University Press 51/1(1995-02), 137-147 0024-6107 51:1<137 1995 51 jlms https://doi.org/10.1112/jlms/51.1.137 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1112/jlms/51.1.137 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Saxon S.A. Department of Mathematics, University of Florida PO Box 118000, Gainesville, Florida 32611-8000, USA E-mail: saxon@math.ufl.edu aut NATIONALLICENCE 700 1- Sánchez Ruiz L.M. EUITI-Departamento de Matemática Aplicada, Universidad Politécnica de Valencia 46071 Valencia, Spain aut NATIONALLICENCE 773 0- Journal of the London Mathematical Society Oxford University Press 51/1(1995-02), 137-147 0024-6107 51:1<137 1995 51 jlms Metadata rights reserved CC BY-NC-4.0 http://creativecommons.org/licenses/by-nc/4.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-oxford