A Generalization of the Alexandrov & Path Topologies of Spacetime via Linear Orders

Let X be a topological space equipped with a collection P of continuous paths. Associated to the pair (X,P) we define a path topology that generalizes the P-topology of Hawking et al. [1] Moreover, we axiomatize a collection P that generates an order that we call an NRT full order. This in turn generalizes the chronology order of spacetime defined via timelike paths. We also investigate the associated Alexandrov topology using ideas of Penrose and Kronheimer[2]. We conclude by defining the notion of a strongly path ordered space (X, τ, P ) that generalizes the concept of strongly causal spacetimes and prove an equivalence involving these topologies.