 A way out of this is to augment \(S\) at each time so that its cardinality matches the highest cardinality that \(S\) ever achieves. Of course, one hopes to do so in a way that is not completely ad hoc. For example, in the context of SDMI, Bacciagaluppi, Donald and Vermaas (1995) show that the “trajectory” through Hilbert space of the spectral components of the reduced state of a physical system will, under reasonable conditions, be continuous, or have only isolated discontinuities, so that the trajectory can be naturally extended to a continuous trajectory (see also Donald 1998). This result suggests a natural family of maps as discussed above, which map each spectral component at one time to its unique continuous evolved component at later times.