# On topological groups containing a Fr\'echet-Urysohn fan

## TL;DR

This paper investigates the structure of topological groups containing a Frechet-Urysohn fan, showing that under certain normality and sequentiality conditions, such groups are either metrizable or have a specific product structure.

## Contribution

It establishes conditions under which topological groups containing a Frechet-Urysohn fan are either metrizable or decomposable into a product of a $k_ ext{omega}$-space and a discrete space.

## Key findings

- Closed metrizable subsets in such groups are locally compact.
- Groups with a direct limit structure are either metrizable or a product of a $k_ ext{omega}$-space and a discrete space.

## Abstract

Suppose G is a topological group containing a (closed) topological copy of the Frechet-Urysohn fan. If G is a perfectly normal sequential space (a normal k-space) then every closed metrizable subset in $G$ is locally compact. Applying this result to topological groups whose underlying topological space can be written as a direct limit of a sequence of closed metrizable subsets, we get that every such a group either is metrizable or is homeomorphic to the product of a $k_\omega$-space and a discrete space.