On topological properties of Fréchet locally convex spaces with the weak topology

S. S. Gabriyelyan, J. Kakol, A. Kubzdela, M. Lopez-Pellicer

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13 Scopus citations

Abstract

We describe the topology of any cosmic space and any ℵ0-space in terms of special bases defined by partially ordered sets. Using this description we show that a Baire cosmic group is metrizable. Next, we study those locally convex spaces (lcs) E which under the weak topology σ(E, E') are ℵ0-spaces. For a metrizable and complete lcs E not containing (an isomorphic copy of) ℓ1 and satisfying the Heinrich density condition we prove that (E, σ(E, E')) is an ℵ0-space if and only if the strong dual of E is separable. In particular, if a Banach space E does not contain ℓ1, then (E, σ(E, E')) is an ℵ0-space if and only if E' is separable. The last part of the paper studies the question: Which spaces (E, σ(E, E')) are ℵ0-spaces? We extend, among the others, Michael's results by showing: If E is a metrizable lcs or a (DF)-space whose strong dual E' is separable, then (E, σ(E, E')) is an ℵ0-space. Supplementing an old result of Corson we show that, for a Čech-complete Lindelöf space X the following are equivalent: (a) X is Polish, (b) Cc(X) is cosmic in the weak topology, (c) the weak*-dual of Cc(X) is an ℵ0-space.

Original languageEnglish
Pages (from-to)123-137
Number of pages15
JournalTopology and its Applications
Volume192
DOIs
StatePublished - 1 Sep 2015

Keywords

  • Banach space
  • K-network
  • Locally convex Fréchet space
  • Weak topology
  • ℵ-space

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