关于Zeeman提出的两个拓扑
On Two Topologies that were suggested by Zeeman
摘要：
在相对论的框架中，时空上的塞曼拓扑类被认为是一个强有力的直观证明，它满足一系列具有物理意义的性质，例如在这种拓扑下的同胚群同构于闵可夫斯基时空中的洛伦兹群和置换，以及任何弯曲时空中的位似对称群。在这篇文章中，我们专注于两个不同的拓扑，塞曼建议作为替代他的精细拓扑，显示它们与两个订单的连接：类时和（非因果）类空的。对于（非因果）类空序，我们引入了零锥的一个划分，它从划分超平面的选择中不变地给出了所需的拓扑。特别是，我们观察到，这两个订单诱导拓扑类的塞曼拓扑，而两个建议的拓扑塞曼自己是交叉拓扑这两个订单拓扑（分别）与流形拓扑。最后，我们列出了一系列的开放性问题和讨论，比较了有界的拓扑和无界的开集的拓扑，以及它们可能的物理解释。
英文摘要：
The class of Zeeman topologies on spacetimes in the frame of relativity theory is considered to be of powerful intuitive justification, satisfying a sequence of properties with physical meaning, such as the group of homeomorphisms under such a topology is isomorphic to the Lorentz group and dilatations, in Minkowski spacetime, and to the group of homothetic symmetries in any curved spacetime. In this article we focus on two distinct topologies that were suggested by Zeeman as alternatives to his Fine topology, showing their connection with two orders: a timelike and a (non-causal) spacelike one. For the (non-causal) spacelike order, we introduce a partition of the null cone which gives the desired topology invariantly from the choice of the hyperplane of partition. In particular, we observe that these two orders induce topologies within the class of Zeeman topologies, while the two suggested topologies by Zeeman himself are intersection topologies of these two order topologies (respectively) with the manifold topology. We end up with a list of open questions and a discussion, comparing the topologies with bounded against those with unbounded open sets and their possible physical interpretation.
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