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.

中文翻译：

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拓扑群的长界线 II：自由群和向量空间

摘要 自由拓扑群和自由阿贝尔拓扑群在空间上的拓扑性质自 1940 年代以来得到了深入研究。在本文中，我们研究了 X 上的自由拓扑 R 向量空间 V ( X )。我们证明了 V ( X ) 是 [ − 1 , 1 ] × X 上的自由阿贝尔拓扑群的商，并用它来证明在赝紧空间上自由拓扑群的现有结果的拓扑向量空间类似物。作为一个应用，我们表明 V ( X ) 的某些子空间族满足作者先前工作中定义的所谓的代数共限性质。