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

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 R-vector space V(X) on X. We show that V(X) is a quotient of the free abelian topological group on [−1,1]×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 algebraic colimit property defined in the authors' previous work.