On sliced spaces: Global Hyperbolicity revisited

We give a topological condition for a generic sliced space to be globally hyperbolic, without any hypothesis on the lapse function, shift function and spatial metric.

Throughout the paper, we will consider I = R.
The author in [3] considered sliced spaces with uniformly bounded lapse, shift and spatial metric; by this hypothesis, it is ensured that parametre t measures up to a positive factor bounded (below and above) the time along the normals to spacelike slices M t , the g t norm of the shift vector β is uniformly bounded by a number and the time-dependent metric g ij dx i dx j is uniformly bounded (below and above) for all t ∈ I(= R), respectively.
Given the above hypothesis, in the same article the following theorem is proved. (Cotsakis). Let (V, g) be a sliced space with uniformly bounded lapse N, shift β and spatial metric g t . Then, the following are equivalent: 1. (M 0 , γ) a complete Riemannian manifold.
In this article we review global hyperbolicity of sliced spaces, in terms of the product topology defined on the space M × R, for some finite dimensional smooth manifold M. We remind the Alexandrov topology T A (see [4]) has a base consisting of open sets of the where ≪ is the chronological order defined as x ≪ y iff there exists a future oriented timelike curve, joining x with y. By J + (x) one denotes the topological closure of I + (x) and by J − (y) that one of I − (y).
We use the definition of global hyperbolicity from [4] where the reader can read about global causality conditions in more detail as well as characterisations for strong causality. In particular, a spacetime is strongly causal, iff it possesses no closed timelike curves and global hyperbolicity is an important causal condition in a spacetime related to major problems such as spacetime singularities, cosmic cencorship etc.
Definition 2.1. A spacetime is globally hyperbolic, iff it is strongly causal and the "causal We prove the following theorem.
Theorem 2.1. Let (V, g) be a Hausdorff sliced space. Then, the following are equivalent.

T
Proof. 2. implies 3. is obvious and that 3. implies 1. can be found in [4]. For 1. implies 2. we consider two events X, Y ∈ V , such that X = Y ; we note that each X ∈ V has two coordinates, say (x 1 , x 2 ), where x 1 ∈ M and x 2 ∈ R. Obviously,  (a, b). Thus, < X, Y >∈ T P . For 2. implies 1. we consider ǫ > 0, such that B h ǫ (A) ∈ T M , so that B h ǫ (A) × (a, b) = B ∈ T P . We let strong causality hold at an event P and consider P ∈ B ∈ T P . We show that there exists < X, Y >∈ T A , such that P ∈< X, Y >⊂ B. Now, consider a simple region R in < X, Y > which contains P and P ∈ Q, where Q is a causally convex-open subset of R.
Thus, we have U, V ∈ Q, such that P ∈< U, V >⊂ Q. Finally, P ∈< U, V >⊂ Q ⊂ B and this completes the proof. Proof. Given the proof of Theorem 2.1, strong causality in V holds iff T P = T A and given Nash's theorem, the closure of B h ǫ (x) × (a, b) will be compact.
We note that neither in Theorem 2.1 nor in Theorem 3.1 we made any hypothesis on the lapse function, shift function or on the spatial metric.