In definition 3 the closure of J − − ( p ) ∩ ∩ J + ( q ) seems strong (in fact, the closures of the sets J ± ± ( p ) imply causal simplicity, the level of the causal hierarchy of spacetimes which stays just below global hyperbolicity). It is possible to remedy this problem strengthening the causality condition as in definition 4 proposed by Minguzzi in 2009. This version clarifies that global hyperbolicity sets a compatibility condition between the causal relation and the notion of compactness: every causal diamond is contained in a compact set and every inextendible causal curve escapes compact sets. Observe that the larger the family of compact sets the easier for causal diamonds to be contained on some compact set but the harder for causal curves to escape compact sets. Thus global hyperbolicity sets a balance on the abundance of compact sets in relation to the causal structure. Since finer topologies have less compact sets we can also say that the balance is on the number of open sets given the causal relation. Definition 4 is also robust under perturbations of the metric (which in principle could introduce closed causal curves). In fact using this version it has been shown that global hyperbolicity is stable under metric perturbations.