  In this paper it is proved that near a compact, invariant, proper subset of a
continuous flow on a compact, connected metric space, at least one, out of
twenty eight relevant dynamical phenomena, will necessarily occur. This result
shows that assuming the connectedness of the phase space implies the existence
of a considerably deeper classification of topological flow behaviour, in the
vicinity of compact invariant sets, than that described in the classical
theorems of Ura-Kimura and Bhatia. The proposed classification brings to light,
in a systematic way, the possibility of occurrence of orbits of infinite height
arbitrarily near the compact invariant in question, and this under relatively
simple conditions. Singularities of smooth vector fields displaying this
strange phenomenon occur in every dimension greater than 2 (in this paper, a
smooth flow on the 3-dimensional sphere exhibiting such an equilibrium is
constructed). Near periodic orbits, the same phenomenon is observable already
in dimension 4 (and on every manifold of dimension greater than 4). As a
corollary to the main result, an elegant characterization of the topological
Hausdorff structure of the set of all compact minimal sets of the flow is
obtained (Theorem 2).
  Keywords: topological behaviour of C0 flows, compact invariant sets, compact
minimal sets, topological Hausdorff structure, non-hyperbolic singularities and
periodic orbits, orbits of infinite height.
