A NEW TOPOLOGY ON SPACE-TIME

A new topology, P approximately, is suggested on space-time definable solely in terms of the causal structure, thus combining ideas of both Zeeman (and more specifically Hawking, King, and McCarthy) and Alexandrov to create a very "physical" topology. It is proved that the distinguishing condition holds on a manifold if and only if this new topology is equivalent to the path topology of Hawking, King, and McCarthy. It is also found that the homeomorphism group of P approximately is the group of chronological and antichronological bijections in the case that the chronology condition holds on the space-time manifold. Thus, if the chronology condition holds P approximately determines the causal structure, and if the distinguishing condition holds P approximately determines the causal, differential, and conformal structures of space-time.