Globally Hyperbolic Spacetimes as Posets

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 $M$ 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 $M$ is a bicontinuous poset whose the interval topology is the manifold topology. Furthermore, we show that, on a causally simple spacetime $M$ as a poset, the global hyperbolicity condition is strictly stronger than the DS-FI cluster point condition.