  We introduce the $R$ cut-off covering spectrum and the cut-off covering
spectrum of a complete length space or Riemannian manifold. The spectra measure
the sizes of localized holes in the space and are defined using covering spaces
called $\delta$ covers and $R$ cut-off $\delta$ covers. They are investigated
using $\delta$ homotopies which are homotopies via grids whose squares are
mapped into balls of radius $\delta$.
  On locally compact spaces, we prove that these new spectra are subsets of the
closure of the length spectrum. We prove the $R$ cut-off covering spectrum is
almost continuous with respect to the pointed Gromov-Hausdorff convergence of
spaces and that the cut-off covering spectrum is also relatively well behaved.
This is not true of the covering spectrum defined in our earlier work which was
shown to be well behaved on compact spaces. We close by analyzing these spectra
on Riemannian manifolds with lower bounds on their sectional and Ricci
curvature and their limit spaces.
