  The Conley index theory is a powerful topological tool for obtaining
information about invariant sets in continuous dynamical systems. A key feature
of Conley theory is that the index is robust under perturbation; given a
continuous family of flows $\{\varphi_\lambda\}$, the index remains constant
over a range of parameter values, avoiding many of the complications associated
with bifurcations.
  This theory is well-developed for flows and homomorphisms, and has even been
extended to certain classes of semiflows. However, in recent years
mathematicians and scientists have become interested in differential
inclusions, often called Filippov systems, and Conley theory has not yet been
developed in this setting. This paper aims to begin extending the index theory
to this larger class of dynamical systems. In particular, we introduce a notion
of perturbation that is suitable for this setting and show that isolating
neighborhoods, a fundamental object in Conley index theory, are stable under
this sense of perturbation. We also discuss how perturbation in this sense
allows us to provide rigorous results about "nearby smooth systems" by
analyzing the discontinuous Filippov system.
