Topological properties of inductive limits of closed towers of metrizable groups

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.