论文标题
拓扑的模型理论
A model theory of topology
论文作者
论文摘要
在麦肯锡和塔斯基的一篇古典论文中，已经提出了拓扑概念的代数化。但是，在麦肯锡和塔斯基的设定中，同构的模型理论概念与连续性的概念不符。我们注意到，如果相反，如果我们考虑由$ a \ sqsubseteq b $定义的预订关系$ \ sqsubseteq $，如果$ a $包含在$ b $的拓扑结束中，则应对应。 专业POSET是部分订购的集合，并具有进一步的更粗糙的预订关系$ \ sqsubseteq $。我们表明，每个专业POSET可以嵌入与某些拓扑空间自然相关的专业poset中，在该拓扑空间中，该顺序关系对应于设定的理论包含。专业半静脉曲张以类似的方式定义，并证明相应的嵌入定理。在这个显然非常薄弱的环境中，回收了一些基本的拓扑事实和概念。这些结构的兴趣源于以下事实：它们也出现在许多相当不同的环境中，甚至与拓扑相距甚远。
An algebraization of the notion of topology has been proposed more than seventy years ago in a classical paper by McKinsey and Tarski. However, in McKinsey and Tarski's setting the model theoretical notion of homomorphism does not correspond to the notion of continuity. We notice that the two notions correspond if instead we consider a preorder relation $ \sqsubseteq $ defined by $a \sqsubseteq b$ if $a$ is contained in the topological closure of $b$. A specialization poset is a partially ordered set endowed with a further coarser preorder relation $ \sqsubseteq $. We show that every specialization poset can be embedded in the specialization poset naturally associated to some topological space, where the order relation corresponds to set-theoretical inclusion. Specialization semilattices are defined in an analogous way and the corresponding embedding theorem is proved. Some basic topological facts and notions are recovered in this apparently very weak setting. The interest of these structures arises from the fact that they also occur in many rather disparate settings, even far removed from topology.
