First order topological structures and theories
First order topological structures and theories
复制标题
一阶拓扑结构和理论
DOI：
--
复制
发表时间：
1987
期刊：
影响因子：
--
通讯作者：
A. Pillay
中科院分区：
文献类型：
--
作者：
A. Pillay
在本文中，我们引入了一阶拓扑结构的概念，并考虑了这种结构中可定义集合的复杂性的各种可能条件，并得出了其几个结果。我们的目标是为有限类别的不稳定理论开发与稳定理论类似的结果。这种努力的“物质基础”是实数领域和复数领域之间的类比，前者是“表现良好”的不稳定结构，后者是典型的稳定结构。从这个意义上说，我们在这里尝试将我们的工作放在一般的拓扑背景下的最小结构[PS]上。但请注意，p 进数和其中可定义的结构也适合我们的分析。在本节的其余部分中，我们讨论从理论上研究拓扑结构模型的几种方法。最终，我们确定了结构的概念，其中拓扑在 Flum 和 Ziegler [FZ] 的意义上是“明确可定义的”。在第 2 节中，我们引入了这样的假设：每个可定义集合都是可定义开集的布尔组合。在第 3 节中，我们引入了（封闭）可定义集的“维度等级”。在第 4 节中，我们考虑定义该等级的结构，并且每个可定义集也具有有限数量的可定义连接的可定义组件。我们证明了在这种条件下存在集合上的素数模型。
In this paper we introduce the notion of a first order topological structure, and consider various possible conditions on the complexity of the definable sets in such a structure, drawing several consequences thereof. Our aim is to develop, for a restricted class of unstable theories, results analogous to those for stable theories. The “material basis” for such an endeavor is the analogy between the field of real numbers and the field of complex numbers, the former being a “nicely behaved” unstable structure and the latter the archetypal stable structure. In this sense we try here to situate our work on o-minimal structures [PS] in a general topological context. Note, however, that the p-adic numbers, and structures definable therein, will also fit into our analysis. In the remainder of this section we discuss several ways of studying topological structures model-theoretically. Eventually we fix on the notion of a structure in which the topology is “explicitly definable” in the sense of Flum and Ziegler [FZ]. In §2 we introduce the hypothesis that every definable set is a Boolean combination of definable open sets. In §3 we introduce a “dimension rank” on (closed) definable sets. In §4 we consider structures on which this rank is defined, and for which also every definable set has a finite number of definably connected definable components. We show that prime models over sets exist under such conditions.