  We introduce two tools, dynamical thickening and flow selectors, to overcome
the infamous discontinuity of the gradient flow endpoint map near
non-degenerate critical points. More precisely, we interpret the stable
fibrations of certain Conley pairs $(N,L)$, established in [2,3], as a
dynamical thickening of the stable manifold. As a first application and to
illustrate efficiency of the concept we reprove a fundamental theorem of
classical Morse theory, Milnor's homotopical cell attachment theorem [1].
Dynamical thickening leads to a conceptually simple and short proof.
