  Utilizing the discrete homotopy methods developed for uniform spaces by
Berestovskii-Plaut, we define the critical spectrum Cr(X) of a metric space,
generalizing to the non-geodesic case the covering spectrum defined by
Sormani-Wei and the homotopy critical spectrum defined by Plaut-Wilkins. If X
is geodesic, Cr(X) is the same as the homotopy critical spectrum, which differs
from the covering spectrum by a factor of 3/2. The latter two spectra are known
to be discrete for compact geodesic spaces, and correspond to the values at
which certain special covering maps, called delta-covers (Sormani-Wei) or
epsilon-covers (Plaut-Wilkins), change equivalence type. In this paper we
initiate the study of these ideas for non-geodesic spaces, motivated by the
need to understand the extent to which the accompanying covering maps are
topological invariants. We show that discreteness of the critical spectrum for
general metric spaces can fail in several ways, which we classify. The
"newcomer" critical values for compact, non-geodesic spaces are completely
determined by the homotopy critical values and refinement critical values, the
latter of which can, in many cases, be removed by changing the metric in a
bi-Lipschitz way.
