Abstract The free topological vector space V ( X ) over a Tychonoff space X is a pair consisting of a topological vector space V ( X ) and a continuous mapping i = i X : X → V ( X ) such that every continuous mapping f from X to a topological vector space E gives rise to a unique continuous linear operator f ‾ : V ( X ) → E with f = f ‾ ∘ i . In this paper, the k-property, Frechet-Urysohn property, κ-Frechet-Urysohn property and countable tightness of free topological vector space over some class of generalized metric spaces are studied. First, we mainly discuss the characterization of a space X such that V ( X ) or the third level of V ( X ) is Frechet-Urysohn or κ-Frechet-Urysohn, respectively. Then we give the characterization of a space X such that the second level of V ( X ) is of countable tightness or is a k-space, respectively.

中文翻译：

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自由拓扑向量空间的注解

摘要 Tychonoff 空间 X 上的自由拓扑向量空间 V ( X ) 是由拓扑向量空间 V ( X ) 和连续映射 i = i X : X → V ( X ) 组成的对，使得每个连续映射 f 从X 到拓扑向量空间 E 产生唯一的连续线性算子 f ‾ : V ( X ) → E ，其中 f = f ‾ ∘ i 。本文研究了一类广义度量空间上自由拓扑向量空间的k-性质、Frechet-Urysohn性质、κ-Frechet-Urysohn性质和可数紧度。首先，我们主要讨论空间 X 的表征，使得 V ( X ) 或 V ( X ) 的第三级分别是 Frechet-Urysohn 或 κ-Frechet-Urysohn。然后我们给出空间 X 的特征，使得 V ( X ) 的第二级分别具有可数紧度或 k 空间。