Long colimits of topological groups II: Free groups and vector spaces

TL;DR

This paper explores the topological structure of free topological vector spaces, establishing their relation to free abelian groups and demonstrating new properties for pseudocompact spaces.

Contribution

It introduces a quotient relationship between free topological vector spaces and free abelian groups, extending known results to new classes of spaces.

Findings

$V(X)$ is a quotient of the free abelian topological group on $[-1,1]\times X$

Topological vector space analogues of free topological group results are established

Certain subspace families of $V(X)$ satisfy the algebraic colimit property

Abstract

Topological properties of the free topological group and the free abelian topological group on a space have been thoroughly studied since the 1940s. In this paper, we study the free topological $\mathbb{R}$-vector space $V(X)$ on $X$. We show that $V(X)$ is a quotient of the free abelian topological group on $[-1,1]\times X$, and use this to prove topological vector space analogues of existing results for free topological groups on pseudocompact spaces. As an application, we show that certain families of subspaces of $V(X)$ satisfy the so-called $\textit{algebraic colimit property}$ defined in the authors' previous work.