Geodesic connectedness of the causal completion in the Lorentzian geometry

We consider the topology of the BS causal completion, and show that it is not distorted in the causal direction. Using this result, we show that the causal completion with spacelike causal boundary in the sense of the BS construction satisfies the formal definition of the global hyperbolicity. We also show that any two causally related points in the causal completion can be connected by a causal geodesic.