The structure of the -ideal of -porous sets
The structure of the -ideal of -porous sets
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理想多孔集的结构
DOI：
10.2307/44153040
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发表时间：
2004
影响因子：
1.7
通讯作者：
Miroslav ZelenJan Pelant
中科院分区：
文献类型：
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作者：
Miroslav ZelenJan Pelant
给出了完备度量空间中非σ多孔集的一般构造方法。这种方法使我们能够回答几个未解决的问题。证明了拓扑完备度量空间的每个非σ-多孔Suslin子集都包含一个非σ-多孔闭子集。我们还给出了一个充分条件，它给出了一个紧集系统包含非σ-多孔元的充分条件。也就是说，如果我们用K(E)表示具有Vietoris拓扑的紧度量空间E的所有紧子集的空间，则证明了包含E的所有可数紧子集的K(E)的每个解析子集必然包含一个元，它是E的非σ多孔子集。我们给出了这一结果在实分析和调和分析中的几个应用(例如三角级数的闭的非σ多孔唯一集的存在性)。最后，我们还研究了紧σ-多孔集的σ-理想的刻画性质。
We show a general method of construction of non-σ-porous sets in complete metric spaces. This method enables us to answer several open questions. We prove that each non-σ-porous Suslin subset of a topologically complete metric space contains a non- σ-porous closed subset. We show also a sufficient condition, which gives that a certain system of compact sets contains a non-σ-porous element. Namely, if we denote the space of all compact subsets of a compact metric space E with the Vietoris topology by K(E), then it is shown that each analytic subset of K(E) containing all countable compact subsets of E contains necessarily an element, which is a non-σ-porous subset of E. We show several applications of this result to problems from real and harmonic analysis (e.g. the existence of a closed non-σ-porous set of uniqueness for trigonometric series). Finally we investigate also descriptive properties of the σ-ideal of compact σ-porous sets.