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Optimal Cardinals for Metrizable Barrelled Spaces
Author(s) -
Saxon S.A.,
Sánchez Ruiz L.M.
Publication year - 1995
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/51.1.137
Subject(s) - metrization theorem , mathematics , combinatorics , bounded function , countable set , space (punctuation) , bounding overwatch , regular polygon , dimension (graph theory) , discrete mathematics , computer science , separable space , mathematical analysis , geometry , artificial intelligence , operating system
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 ℵ 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 ℵ 1 ⩽ 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 (ℵ 1 = c ). The reconstruction is vital in the characterizations of b and in subsequent papers answering open questions about countable enlargements.