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Countability, Completeness and the Closed Graph Theorem

  • R. Beattie
  • H.-P. Butzmann

Abstract

The webs of M. De Wilde [4] have made an enormous contribution to the closed graph theorems in locally convex spaces(lcs). Although webs have a very intricate layered construction, two properties in particular have contributed to the closed graph theorem. First of all, webs possess a strong countability condition in the range space which suitably matches the Baire property of Fréchet spaces in the domain space; as a result the zero neighbourhood filter is mapped to a p-Cauchy filter, a filter attempting to settle down. Secondly webs provide a completeness condition which allow p-Cauchy filters to converge.

Keywords

Countable Basis Closed Graph Graph Theorem Completeness Property Closed Graph Theorem 
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References

  1. 1.
    R. Beattie, Convergence spaces with webs, Math. Nachr. 116, 159–164 (1984).MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    R. Beattie, A convenient category for the closed graph theorem, Categorical Topology, Proc. Conference Toledo, Ohio 1983, Heldermann, Berlin, 29–45 (1984).Google Scholar
  3. 3.
    R. Beattie and H. -P. Butzmann, Strongly first countable convergence spaces, Convergence Structures 1984, Proc. Conference on Convergence, Bechyne, Czechoslovakia, Akademie-Verlag, Berlin, 39–46 (1985).Google Scholar
  4. 4.
    M. De Wilde, Closed Graph Theorems and Webbed Spaces, Research Notes in Mathematics 19, Pitman, London (1978).Google Scholar
  5. 5.
    H. Jarchow, “Locally Convex Spaces”, Teubner, Stuttgart (1981).MATHGoogle Scholar
  6. 6.
    W. Robertson, On the closed graph theorem and spaces with webs, Proc. London Math. Soc. (3) 24, 692–738 (1972).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1988

Authors and Affiliations

  • R. Beattie
    • 1
  • H.-P. Butzmann
    • 2
  1. 1.Dept. of Mathematics and Computer ScienceMount Allison UniversitySackvilleCanada
  2. 2.Fakultät für Mathematik und InformatikUniversität MannheimMannheimDeutschland

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