Global Hyperbolicity in Space-Time Manifold

Global hyperbolicity is the most important condition on causal structure space-time, which is involved in problems as cosmic censorship, predictability etc. An open set O is said to be globally hyperbolic if, i) for every pair of points x and y in O the intersection of the future of x and the past of y has compact closure i.e., a space-time is said to be globally hyperbolic if the sets are compact for all (i.e., no naked singularity can exist in space-time topology), and ii) strong causality holds on O i.e., there are no closed or almost closed time like curves contained in O. Here is causal future and is the causal past of an event x. If a space-time is timelike or null geodesically incomplete but cannot be embedded in a larger space-time then we say that it has a singularity. An attempt is taken here to discuss global hyperbolicity and space-time singularity by introducing definitions, propositions and displaying diagrams appropriately