On topological groups with a small base and metrizability

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28 Scopus citations

Abstract

A (Hausdorff) topological group is said to have a φ-base if it admits a base of neighbourhoods of the unit, {Uα : α ε NN}, such that Uα ⊂ Uβ whenever β ≤ α for all α, β ε NN. The class of all metrizable topological groups is a proper subclass of the class TGφ of all topological groups having a φ-base. We prove that a topological group is metrizable iff it is Fréchet-Urysohn and has a φ-base. We also show that any precompact set in a topological group G ε TGG is metrizable, and hence G is strictly angelic. We deduce from this result that an almost metrizable group is metrizable iff it has a φ-base. Characterizations of metrizability of topological vector spaces, in particular of Cc(X), are given using φ-bases. We prove that if X is a submetrizable kω-space, then the free abelian topological group A(X) and the free locally convex topological space L(X) have a φ-base. Another class TGCR of topological groups with a compact resolution swallowing compact sets appears naturally. We show that TGCR and TGG are in some sense dual to each other. We conclude with a dozen open questions and various (counter)examples.

Original languageEnglish
Pages (from-to)129-158
Number of pages30
JournalFundamenta Mathematicae
Volume229
Issue number2
DOIs
StatePublished - 1 Jan 2015

Keywords

  • Dual group
  • Fréchet-Urysohn space
  • Locally convex space
  • Metrizable group
  • Topological group

ASJC Scopus subject areas

  • Algebra and Number Theory

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