## Abstract

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 G_{n} is a closed subgroup of G, (3) if K is a compact subset of G, then K⊆ G_{m} 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 G_{n} is open in G_{n}_{+}_{1} for every n≥ m, so G is metrizable, or (ii) all groups G_{n} are locally compact and G is a sequential non-Fréchet–Urysohn space.

Original language | English |
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Article number | 33 |

Journal | Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales - Serie A: Matematicas |

Volume | 116 |

Issue number | 1 |

DOIs | |

State | Published - 1 Jan 2022 |

## Keywords

- 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