Consider a set $M$ equipped with a structure $*$. We call a natural topology $T_*$, on $(M,*)$, the topology induced by $*$. For example, a natural topology for a metric space $(X,d)$ is a topology $T_d$ induced by the metric $d$ and for a linearly ordered set $(X,<)$ a natural topology should be the topology $T_<$ that is induced by the order $<$. This fundamental property, for a topology to be called "natural", has been largely ignored while studying topological properties of spacetime manifolds $(M,g)$ where $g$ is the Lorentz "metric", and the manifold topology $T_M$ has been used as a natural topology, ignoring the spacetime "metric" $g$. In this survey we review critically candidate topologies for a relativistic spacetime manifold, we pose open questions and conjectures with the aim to establish a complete guide on the latest results in the field, and give the foundations for future discussions. We discuss the criticism against the manifold topology, a criticism that was initiated by people like Zeeman, Göbel, Hawking-King-McCarthy and others, and we examine what should be meant by the term "natural topology" for a spacetime. Since the common criticism against spacetime topologies, other than the manifold topology, claims that there has not been established yet a physical theory to justify such topologies, we give examples of seemingly physical phenomena, under the manifold topology, which are actually purely effects depending on the choice of the topology; the Limit Curve Theorem, which is linked to singularity theorems in general relativity, and the Theorem of Gao-Wald type of "time dilation" are such examples. }
在固定流形上的洛伦兹度规空间中，有两类拓扑最常被放置。正如我对R. Geroch [Relativity，259（1970）; Gen. Rel.重力，2，61（1971）]，然而，这些标准类都没有正确地捕捉到全局时空相似性的概念。特别是，Geroch提出的例子来说明，一个，紧凑的开放拓扑，一般似乎太粗糙，而另一个，开放（惠特尼）拓扑，一般似乎太精细。在进一步阐述了这些失败的数学和物理原因之后，我构建了一个拓扑结构，成功地捕捉了全局时空相似性的概念，并研究了它的一些数学和物理性质。
Geodesic geometry of 2+1-D Dirac materials subject to artificial, quenched gravitational singularities
Condensed Matter Theory Center and Joint Quantum Institute, Department of Physics, University of Maryland（马里兰大学凝聚态理论中心与联合量子研究所）；Department of Physics and Astronomy and Rice Center for Quantum Materials, Rice University（莱斯大学物理与天文学系及量子材料中心）
