Free locally convex spaces with a small base

Saak Gabriyelyan, Jerzy Ka̧kol

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


The paper studies the free locally convex space L(X) over a Tychonoff space X. Since for infinite X the space L(X) is never metrizable (even not Fréchet-Urysohn), a possible applicable generalized metric property for L(X) is welcome. We propose a concept (essentially weaker than first-countability) which is known under the name a G-base. A space X has a G-base if for every x∈ X there is a base { Uα: α∈ NN} of neighborhoods at x such that Uβ⊆ Uα whenever α≤ β for all α, β∈ NN, where α= (α(n)) n N≤ β= (β(n)) n N if α(n) ≤ β(n) for all n∈ N. We show that if X is an Ascoli σ-compact space, then L(X) has a G-base if and only if X admits an Ascoli uniformity U with a G-base. We prove that if X is a σ-compact Ascoli space of NN-uniformly compact type, then L(X) has a G-base. As an application we show: (1) if X is a metrizable space, then L(X) has a G-base if and only if X is σ-compact, and (2) if X is a countable Ascoli space, then L(X) has a G-base if and only if X has a G-base.

Original languageEnglish
Pages (from-to)575-585
Number of pages11
JournalRevista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Issue number2
StatePublished - 1 Apr 2017


  • C(X)
  • Compact resolution
  • Free locally convex space
  • G-base

ASJC Scopus subject areas

  • Analysis
  • Algebra and Number Theory
  • Geometry and Topology
  • Computational Mathematics
  • Applied Mathematics


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