As mentioned, in older literature, the condition of causality in the first and third definitions of global hyperbolicity given above is replaced by the stronger condition of strong causality. In 2007, Bernal and Sánchez showed that the condition of strong causality can be replaced by causality. In particular, any globally hyperbolic manifold as defined in 3 is strongly causal. Later Hounnonkpe and Minguzzi proved that for quite reasonable spacetimes, more precisely those of dimension larger than three which are non-compact or non-totally vicious, the 'causal' condition can be dropped from definition 3.