Topological properties of inductive limits of closed towers of metrizable groups

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Let {Gn}n∈ω be a closed tower of metrizable groups. Under a mild condition called (GC) and which is strictly weaker than PTA condition introduced by Shimomura et al. (J Math Kyoto Univ 38:551–578, 1998), we show that: (1) the inductive limit G=g-lim→Gn of the tower is a Hausdorff group, (2) every Gn is a closed subgroup of G, (3) if K is a compact subset of G, then K⊆ Gm for some m∈ ω, (4) G has countable tightness and a G-base, (5) G is an ℵ-space, (6) G is a sequentially Ascoli space if and only if either (i) there is an m∈ ω such that Gn is open in Gn+1 for every n≥ m, so G is metrizable, or (ii) all groups Gn are locally compact and G is a sequential non-Fréchet–Urysohn space.

Original languageEnglish
Article number33
JournalRevista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas
Issue number1
StatePublished - 1 Jan 2022


  • Ascoli
  • Fréchet–Urysohn
  • Inductive limit
  • Metrizable group
  • ℵ-Space

ASJC Scopus subject areas

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


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