  We study mathematical programs with switching constraints (MPSC)from the
topological perspective. Two basic theorems from Morse theory are proved.
Outside the W-stationary point set, continuous defor-mation of lower level sets
can be performed. However, when passing a W-stationary level, the topology of
the lower level set changes via the attachment of a w-dimensional cell. The
dimension w equals the W-index of the nondegenerate W-stationary point. The
W-index depends on both the number of negative eigenvalues of the restricted
Lagrangian's Hessian and the number of bi-active switching constraints. As a
consequence, we show the mountain pass theorem for MPSC. Additionally, we
address the question if the assumption on the nondegeneracy of W-stationary
points is too restrictive in the context of MPSC. It turns out that all
W-stationary points are generically nondegenerate. Besides, we examine the gap
between nondegeneracy andstrong stability of W-stationary points. A complete
characterizationof strong stability for W-stationary points by means of first
and second order information of the MPSC defining functions under linear
independence constraint qualification is provided. In particular, all bi-active
Lagrange multipliers of a strongly stable W-stationary point cannot vanish.
