Cosmological time functions in Lorentzian length spaces

In this paper, we define the concept of a cosmological time function on Lorentzian length spaces. We prove that a Lorentzian length space is globally hyperbolic if and only if the cosmological time function of a Lorentzian length space within its conformal class is regular. Additionally, we establish an ordering in the space of Lorentzian length spaces on a proper metric space and investigate stable causality in its fundamental form. Furthermore, we show that this definition of stable causality implies the existence of a time function inspired by the cosmological time function.