  In this paper we introduce a new compactness condition - Property (C) - for
flows in (not necessary locally compact) metric spaces. For such flows a Conley
type theory can be developed. For example (regular) index pairs always exist
for Property-(C) flows and a Conley index can be defined. An important class of
flows satisfying this compactness condition are LS-flows. We apply E-cohomology
to index pairs of LS-flows and obtain the E-cohomological Conley index. We
formulate a continuation principle for the E-cohomological Conley index and
show that all LS-flows can be continued to LS-gradient flows. We show that the
Morse homology of LS-gradient flows computes the E-cohomological Conley index.
We use Lyapunov functions to define the Morse-Conley-Floer cohomology in this
context, and show that it is also isomorphic to the E-cohomological Conley
index.
