Fr\'{e}chet-Urysohn fans in free topological groups

TL;DR

This paper constructs a Fréchet-Urysohn fan within certain free topological groups, demonstrating that many such groups contain complex subspaces when built from spaces including the rationals.

Contribution

It shows how to embed a Fréchet-Urysohn fan into free topological groups with functorial embeddings, expanding understanding of their topological structure.

Findings

Constructed a Fréchet-Urysohn fan in a topological group with a functorial embedding.

Many free topological groups built from spaces containing the rationals include the fan.

Demonstrated the presence of complex subspaces in various free topological groups.

Abstract

In this paper we answer the question of T. Banakh and M. Zarichnyi constructing a copy of the Fr\'echet-Urysohn fan $S_\w$ in a topological group $G$ admitting a functorial embedding $[0,1]\subset G$. The latter means that each autohomeomorphism of $[0,1]$ extends to a continuous homomorphism of $G$. This implies that many natural free topological group constructions (e.g. the constructions of the Markov free topological group, free abelian topological group, free totally bounded group, free compact group) applied to a Tychonov space $X$ containing a topological copy of the space $\IQ$ of rationals give topological groups containing $S_\w$.