由偏序集索引的Stone空间划分
Stone space partitions indexed by a poset
AI总结：
本文研究由偏序集或PO系统索引的Stone空间修剪划分理论，分析序性质与拓扑性质的关联、划分的扩张完备性、存在充要条件及同胚唯一性，得到Cantor集上满足特定闭包条件的实索引划分推论。
摘要：
满足对所有$p\in P$有$\bar{X_{p}}=\bigcup_{q\leqslant p}X_{q}$这类条件的Stone空间划分$\{X_{p}\mid p\in P\}$（其中$P$是偏序集或PO系统（带特异子集的偏序集）），在本原布尔代数和$\omega$-范畴结构的研究中自然出现。研究这类划分的一个核心概念是$p$-trim（$p$-修剪）开集，它恰好与所有满足$q\geqslant p$的$X_{q}$相交；对于Stone空间而言，这是伪不可分解集的拓扑等价概念。本文发展了由偏序集或PO系统索引、且修剪集构成拓扑邻域基的Stone空间无限划分理论。我们研究了偏序集/PO系统的序性质与划分的拓扑性质之间的相互作用，考察了这类划分的扩张和完备化，并推导了偏序集/PO系统上保证所研究的各类划分存在的充要条件。我们还明确了在给定划分元素的孤立点结构和有界性选择的前提下，带有由指定PO系统索引的修剪划分的第二可数Stone空间在同胚意义下唯一的情形。我们结果的一个推论是：存在Cantor集的一个划分$\{X_{r}\mid r\in[0,1]\}$，使得对所有$r\in[0,1]$有$\bar{X_{r}}=\bigcup_{s\leqslant r}X_{s}$。
英文摘要：
Stone space partitions $\{X_{p}\mid p\in P\}$ satisfying conditions like $\bar{X_{p}}=\bigcup_{q\leqslant p}X_{q}$ for all $p\in P$, where $P$ is a poset or PO system (poset with a distinguished subset), arise naturally in the study both of primitive Boolean algebras and of $ω$-categorical structures. A key concept for studying such partitions is that of a $p$-trim open set which meets precisely those $X_{q}$ for which $q\geqslant p$; for Stone spaces, this is the topological equivalent of a pseudo-indecomposable set. This paper develops the theory of infinite partitions of Stone spaces indexed by a poset or PO system where the trim sets form a neighbourhood base for the topology. We study the interplay between order properties of the poset/PO system and topological properties of the partition, examine extensions and completions of such partitions, and derive necessary and sufficient conditions on the poset/PO system for the existence of the various types of partition studied. We also identify circumstances in which a second countable Stone space with a trim partition indexed by a given PO system is unique up to homeomorphism, subject to choices on the isolated point structure and boundedness of the partition elements. One corollary of our results is that there is a partition $\{X_{r}\mid r\in[0,1]\}$ of the Cantor set such that $\bar{X_{r}}=\bigcup_{s\leqslant r}X_{s}\text{ for all }r\in[0,1]$.
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