The condition of pseudoconvexity has important implications for geodesic structures of space-time. Low showed that this condition guarantees the existence of a smooth structure of the space of geodesics. Recently, the place of this property within the causality ladder is in the attention. There is a conjecture which states that every strongly causal space-time is causally simple if and only if it is null pseudoconvex. In this paper, we study the difference of causal and null pseudoconvexity in space-times and introduce the limit geodesic segment as an equivalence condition to the pseudoconvexity. Then, we show that in the case of open subspaces of n-Minkowski space-time, this is causal geodesic connectedness. Finally, we prove a refined form of the conjecture that says every strongly causal space-time is causally simple if and only if it is maximal null pseudoconvex.

A new characterization for global hyperbolicity is given. The concept of cosmological time function and its regularity was considered by Anderson et al. [“The cosmological time function,” Class. Quantum Grav. 15, 309 (1998)]10.1088/0264-9381/15/2/006 and it was proved that if the cosmological time function of (M, g) is regular then it is globally hyperbolic. In this paper it is proved that if (M, g) is globally hyperbolic then there is a smooth function Ω > 0 such that the cosmological time function of (M, Ωg) is regular. It is also proved that the cosmological time function of Friedman-Robertson-Walker spacetime ((a, b) × f H, −dt2 + f h), a, b < ∞, is regular and in addition the regularity of cosmological time function for this kind of spacetimes is stable in \documentclass[12pt]{minimal}\begin{document}$\rm {Lor}(\it M)$\end{document} Lor (M).

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Using the relation K + , we prove that a certain type of stably causal spacetimes is a jointly bicontinuous poset whose interval topology is the manifold topology.

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