拓扑的一个模型理论
A model theory of topology
AI总结：
本文通过引入由拓扑闭包定义的预序，提出特化偏序集与特化半格，证明其可嵌入拓扑空间，并使模型论同态对应连续性。
摘要：
七十多年前，McKinsey 和 Tarski 在一篇经典论文中提出了拓扑概念的代数化。然而，在 McKinsey 和 Tarski 的框架中，模型论意义上的同态概念并不对应于连续性概念。我们注意到，如果转而考虑由 a \sqsubseteq b（当 a 包含于 b 的拓扑闭包中时）定义的预序关系 $\sqsubseteq$，这两个概念便相互对应。\n\n特化偏序集是一个偏序集，并被赋予另一个更粗的预序关系 $\sqsubseteq$。我们证明，每个特化偏序集都可嵌入某个拓扑空间自然关联的特化偏序集中，其中序关系对应于集合论意义上的包含关系。特化半格以类似方式定义，并证明了相应的嵌入定理。一些基本的拓扑事实与概念在这一看似极弱的框架中得以恢复。这些结构的意义在于，它们也出现在许多相当分散的领域中，甚至远远超出拓扑学的范围。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
