Negligible sets and the reconstruction of spacetime topology

Abstract

When strong causality fails, the Alexandrov topology of a spacetime is strictly coarser than the manifold topology. Since pathologies confined to sets of measure zero are routinely disregarded in relativity, one might expect the deficit to be repairable up to a negligible set; we show that it is not, and determine how much volume must be discarded, as an infimum that need not be attained. Let Let   sigma subseteq tau be topologies on a set X, let I be an ideal on X, and let sigma^* (I) be the topology generated by the differences S \  N with S in sigma and N in I. If  I contains no nonempty tau-open set and tau subseteq sigma^*(I), then tau_{theta} subseteq sigma, where tau_{theta} is the topology of theta-open sets; for regular tau this forces sigma=tau, and the bound is attained. We characterize the admissible ideals: an open set O is the open core of an ideal recovering tau from sigma precisely when O in sigma and the two topologies agree on X \ O. Such refinements preserve the Hausdorff property in both directions, but preserve neither the T₀ nor the T₁ axiom. Consequently the strongly causal set of a spacetime is invariant under refinement by every codense ideal, and the region that must be discarded in order to recover the manifold topology from chronology is a union of chronological diamonds, of volume at least that of the chronology-violating set. For an explicit Lorentzian cylinder the infimum of these volumes is zero and is not attained.