Topological properties of inductive limits of closed towers of mertrizable groups

TL;DR

This paper investigates the topological properties of inductive limits of closed towers of metrizable groups, establishing conditions under which the limit space exhibits desirable features like being Hausdorff, having a G-base, and being an spaces.

Contribution

It introduces a weaker condition than previously known, called (GC), and characterizes the topological structure of inductive limits of such towers, including metrizability and sequential properties.

Findings

The inductive limit is Hausdorff and contains each group as a closed subgroup.

Compact subsets are contained in some group in the tower.

The inductive limit space has a spaces structure and countable tightness.

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.