Mathematics – Logic
Scientific paper
Apr 1977
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1977gregr...8..245b&link_type=abstract
General Relativity and Gravitation, vol. 8, Apr. 1977, p. 245-257.
Mathematics
Logic
2
Metric Space, Space-Time Functions, Topology, Astronomical Models, Causes, Mathematical Models, Relativity
Scientific paper
A metric topology H(barred M) is defined on the causal completion, barred M, of a causally continuous space-time, M, by the requirement that a sequence of points in barred M approach some point, p, of barred M if and only if the chronological pasts and futures of the points approach, respectively, the chronological past and future of p. Two distinct ways of defining metrics on barred M which generate H(barred M) are discussed, one involving the definition of a distance on the collection, C(M), of closed subsets of M, and the other involving the introduction of a bounded measure on M. The extended Alexandrov topology, A(barred M), is also defined on barred M, and it is shown that H(barred M) is at least as coarse as A(barred M), so that each open set of H(barred M) is also an open set in A(barred M). It is demonstrated that H(barred M) is the same as A(barred M) when M admits a compact Cauchy surface, implying that the extended Alexandrov topology is metrizable when M has a compact Cauchy surface. It is suggested that the two topologies on barred M agree for all causally simple space-times.
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