The space of causal curves and separation axioms

Details The space of causal curves and separation axioms The space of causal curves and separation axioms

Jan 2012

adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2012cqgra..29a5014p&link_type=abstract

Classical and Quantum Gravity, Volume 29, Issue 1, pp. 015014 (2012).

Mathematics

Logic

Scientific paper

We consider the space of causal curves of a spacetime with the τ0-topology, introduced by R J Low. He has proved that, for every strongly causal spacetime, the globally hyperbolic condition is equivalent to the Hausdorffness of the τ0-topology. In this paper, we show that, for every distinguishing spacetime, the τ0-topology satisfies the T1-separation axiom, and in the case of chronological spacetime, it satisfies the T0-separation axiom. Then, we define the notions of a ‘proper causal closed curve’ and a ‘somewhere dense causal curve’ and show that the existence of each of these curves implies the existence of the other one. Moreover, the T1-separation axiom is violated if and only if any such curves can be constructed.

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