Topological properties of inductive limits of closed towers of mertrizable groups

Abstract

Let $\{ G_n\}_{n\in\w}$ be a closed tower of metrizable groups. Under a mild condition called $(GC)$ and which is strictly weaker than $PTA$ condition introduced in [22], we show that: (1) the inductive limit $G=\mbox{g-}\underrightarrow{\lim}\, G_n$ 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 $K\subseteq G_m$ for some $m\in\omega$, (4) $G$ has a $\mathfrak{G}$-base and countable tightness, (5) $G$ is an $\aleph$-space, (6) $G$ is an Ascoli space if and only if either (i) there is $m\in\omega$ such that $G_n$ is open in $G_{n+1}$ for every $n\geq m$, so $G$ is metrizable; or (ii) all the groups $G_n$ are locally compact and $G$ is a sequential non-Fr\'{e}chet--Urysohn space.