The representation of spacetime through time functions

The properties of the stable distance over stable spacetimes are used as a reference to propose a simplified, abstract notion of spacetime. Our analysis establishes that the fundamental structures of spacetime, namely its topology, causal order, and (upper semi-continuous) Lorentzian distance, can be derived from a general and minimalistic set of axioms. Specifically, it is shown that spacetime can be represented as nothing more than a family of functions defined over an arbitrary set, the functions being a posteriori interpreted as rushing time functions. The proof makes use of the product trick which reduces causality and metricity to causality in a space with one additional dimension, so leading to a unification for the notions of time function and proper time. Ultimately, our results show that time fully characterizes spacetime.