A note on null distance and causality encoding

Under natural conditions, the null distance introduced by Sormani and Vega (2016 Class. Quantum Grav. 33 085001) is a metric space distance function on spacetime, which, in a certain precise sense, can encode the causality of spacetime. The null distance function requires the choice of a time function. The purpose of this note is to observe that the causality assumptions related to such a choice in results used to establish global encoding of causality, due to Sakovich and Sormani (2023 J. Math. Phys. 64 012502) and to Burtscher and García-Heveling (2022 arXiv:2209.15610 [math.DG]), can be weakened.