Compactness up to breaking needs closedness or a proper compactness package

The compactness theorem Compactness up to breaking of Morse trajectory spaces uses closedness of $M$ in three distinct places. First, completeness of the downward flow, which gives full time-parametrized connecting orbits (their height parametrizations instead have the finite domain $[f(q),f(p)]$); on a compact manifold every smooth vector field is complete (Every smooth vector field on a compact manifold is complete). Second, compactness of $M$ in the Arzelà--Ascoli step: the equicontinuous family of height parametrizations has values in a compact metric target, so its compact-open closure is compact (Under Choice, an equicontinuous family into a compact metric target has compact compact-open closure). Third, finiteness of the critical set in each index, used to split a limiting height map at the finitely many critical points it actually meets and to conclude that it is a finite broken trajectory (A Morse function on a compact manifold has finitely many critical points); indeed the theorem is stated for a Morse--Smale pair in the sense of Morse--Smale pairs, which presupposes a complete field.

On a noncompact manifold one needs a suitable compactness package for the connecting trajectories — properness is one sufficient way to obtain it, for instance the proper smooth functions and compact Morse slabs of Proper smooth functions and compact Morse slabs together with the trapped-trajectory completeness of Proper Morse slabs prevent finite-time escape of connecting trajectories — and exclude escape of trajectories to infinity; completeness of the flow is not automatic (Completeness of a gradient flow is an extra hypothesis on a noncompact manifold), and the slabs give only the conditional nonescape conclusion they state. Without such hypotheses the conclusion can fail, so an assertion of a Morse complex on a noncompact manifold must state the compactness and completeness hypotheses it uses and may not rely on the Morse--Smale condition alone.