  Given an arbitrary continuous flow on a manifold M, let CMin be the set of
its compact minimal sets, endowed with the Hausdorff metric, and S the subset
of those that are Lyapunov stable. A topological characterization of the
interior of S, the set of Lyapunov stable compact minimal sets that are away
from Lyapunov unstable ones is given, together with a description of the
dynamics around it. In particular, int S is locally a Peano continuum (Peano
curve) and each of its countably many connected components admits a complete
geodesic metric.
  This result establishes unexpected connections between the local topology of
CMin and the dynamics of the flow, providing criteria for the local detection
of Lyapunov instability by merely looking at the topology of CMin. For
instance, if CMin is not locally connected at some compact minimal set Q (seen
as a "point" of CMin), then every neighbourhood of Q in M contains Lyapunov
unstable compact minimal sets (hence, if CMin is nowhere locally connected,
then every neighbourhood of each compact minimal set contains infinitely many
Lyapunov unstable compact minimal sets).
