It is well-known that a spacetime with its causal relation is a partially ordered set (poset for short). If it is globally hyperbolic, then it is a bicontinuous poset whose the interval topology is the manifold topology. In this work, we will state a new condition on a poset, which is called DS-FI cluster point condition and we show that when a causally simple spacetime ${\mathscr{M}}$ as a poset with the manifold topology satisfies this condition, then ≪= I+ (where by ≪, we mean the way-below relation on arbitrary poset and by I+, we mean the chronological relation on spacetime), and ${\mathscr{M}}$ is a bicontinuous poset whose the interval topology is the manifold topology. Furthermore, we show that, on a causally simple spacetime ${\mathscr{M}}$ as a poset, the global hyperbolicity condition is strictly stronger than the DS-FI cluster point condition.

中文翻译：

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作为位姿的全局双曲时空

众所周知，具有因果关系的时空是偏序集（简称poset）。如果它是全局双曲线的，那么它是一个双连续的偏序集，其区间拓扑是流形拓扑。在这项工作中，我们将在偏序集上陈述一个新条件，称为 DS-FI 簇点条件，我们证明当因果简单时空 ${\mathscr{M}}$ 作为具有流形拓扑的偏序集满足这个条件，则 ≪= I+（其中 ≪ 表示任意偏序组上的远低于关系，I+ 表示时空上的时间关系），${\mathscr{M}}$ 是一个双连续偏序组，其区间拓扑是流形拓扑。此外，我们表明，在因果简单的时空 ${\mathscr{M}}$ 作为偏序集，