Directed sets and cofinal types
Directed sets and cofinal types
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有向集和共尾类型
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发表时间：
1985
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通讯作者：
S. Todorcevic
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作者：
S. Todorcevic
证明了1，S. @1，Xxxl和[l]<@是唯一的大小为Sl的有向集的共尾类型，但存在许多大小为连续统的有向集的共尾类型。一个偏序集合D是有向的，如果D的每两个元素在D中有一个上界。本文讨论了拓扑学中Moore-Smith收敛理论中有关定向集的一些基本问题[12，3，19，9]。一个这样的问题是确定在给定的空间类中定义闭包算子所需的“所有基本类型的有向集”[3，p. 47]。关于这个问题，J. Tukey提出了以下重要概念[19]。两个有向集D和E是共尾相似的，如果存在一个偏序集C，其中两个有向集D和E都可以嵌入为共尾子集。他表明，这是一个等价关系，并认为D和E是cofilamentally类似的当且仅当有一个收敛映射从D到E，也是一个收敛映射从E到D。这种关系的等价类称为共尾类型。从那时起，这一概念被许多作者广泛研究[4，13，7，8]。从这个概念的第一次介绍开始，已经知道1，，1，X x 1和[X1]<@表示大小<bt 1的有向集的不同共尾类型，但已知的这种类型不超过五种。本文的主要结果表明：1，1，X x 1和[X 1]<@是特征标< 81的空间中唯一的共尾收敛类型，它可以在不附加集合论假设的情况下构造.另一方面，我们将构造许多不同的共尾类型的大小连续统的有向集。这给出了J. Isbell [7]的问题1的解。本文还包含了几个结果的结构类的所有共尾类型，以及一个结果分解任意偏序集到有向集。这篇文章的结果在1982年2月至3月间得到了证明，并于1983年1月提交给ASL。1.分解定理在本节中，我们将证明任意偏序集可以根据其反链的大小分解为许多有向子集。这个结果与F的一个未发表的问题有关。Galvin关于Dilworth分解定理[6]的结果，推广了E.米尔纳和K. Prikry [11].偏序的传递性条件在我们的证明中没有使用，所以我们陈述我们的结果以便适用于任意的。1980年《数学学科分类》。小学03 E05、03 E35;中学06 A10、18 B35、54 A15。C1985美国数学学会0002-9947/85每页$1.00 + $.25
We show that 1, S. @1, X x xl and [l]<@ are the only cofinal types of directed sets of size Sl, but that there exist many cofinal types of directed sets of size continuum. A partially ordered set D is directed if every two elements of D have an upper bound in D. In this note we consider some basic problems concerning directed sets which have their origin in the theory of Moore-Smith convergence in topology [12, 3, 19, 9]. One such problem is to determine "all essential kind of directed sets" needed for defining the closure operator in a given class of spaces [3, p. 47]. Concerning this problem, the following important notion was introduced by J. Tukey [19]. Two directed sets D and E are cofinally similar if there is a partially ordered set C in which both can be embedded as cofinal subsets. He showed that this is an equivalence relation and that D and E are cofinally similar iff there is a convergent map from D into E and also a convergent map from E into D. The equivalence classes of this relation are called cofinal types. This concept has been extensively studied since then by various authors [4, 13, 7, 8]. Already, from the first introduction of this concept, it has been known that 1, , 1, X x 1 and [X1]<@ represent different cofinal types of directed sets of size < btl, but no more than five such types were known. The main result of this paper shows that 1, , 1, X x 1 and [X1]<@ are the only cofinal types of convergence in spaces of character < 81 which can be constructed without additional set-theoretic assumptions. On the other hand, we shall construct many different cofinal types of directed sets of size continuum. This gives a solution to Problem 1 of J. Isbell [7]. The paper also contains several results about the structure of the class of all cofinal types, as well as a result about decomposing arbitrary partially ordered sets into directed sets. The results of this note were proved in February-March 1982 and presented to the ASL in January 1983. 1. A decomposition theorem. In this section we show that an arbitrary partially ordered set can be decomposed into a number of its directed subsets depending on the sizes of its antichains. This result is connected with an unpublished problem of F. Galvin concerning the Dilworth decomposition theorem [6] and it generalizes a similar result of E. Milner and K. Prikry [11]. The transitivity condition of a partial ordering is not used in our proof, so we state our result so as to apply to an arbitrary Received by the editors December 3, 1984. 1980 MathPmaties Subjeet Classification. Primary 03E05, 03E35; Secondary 06A10, 18B35, 54A15. C1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page