Fans及其在General Topology、泛函分析和拓扑代数中的应用
Fans and their applications in General Topology, Functional Analysis and Topological Algebra
摘要：
拓扑空间$X$的闭子集族称为$X$中的（严格）$Cld$ -扇，如果这个族在$X$中是（严格）紧有限的但不是局部有限的。（严格）$Cld$ -fan的应用是基于一个简单的观察，即$k$ -空间不包含$Cld$ -fan，Ascoli空间不包含严格的$Cld$ -fan。在本文中，我们开发的机械（严格）球迷和应用它来检测$k$ -空间和Ascoli性质的空间，自然出现在一般拓扑，泛函分析，拓扑代数。特别是，我们检测（广义）度量空间$X$，其函子空间，函数空间，自由（仿）拓扑（交换）群，自由（局部凸）线性拓扑空间，自由（劳森）拓扑半格，自由（仿）拓扑（Clifford，Abel）逆半群是$k$ -空间或Ascoli空间。
英文摘要：
A family of closed subsets of a topological space $X$ is called a (strict) $Cld$-fan in $X$ if this family is (strictly) compact-finite but not locally finite in $X$. Applications of (strict) $Cld$-fans are based on a simple observation that $k$-spaces contain no $Cld$-fan and Ascoli spaces contain no strict $Cld$-fan. In this paper we develop the machinery of (strict) fans and apply it to detecting the $k$-space and Ascoli properties in spaces that naturally appear in General Topology, Functional Analysis, and Topological Algebra. In particular, we detect (generalized) metric spaces $X$ whose functor-spaces, functions spaces, free (para)topological (abelian) groups, free (locally convex) linear topological spaces, free (Lawson) topological semilattices, and free (para)topological (Clifford, Abelian) inverse semigroups are $k$-spaces or Ascoli spaces.
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