On the transversality conditions and their genericity

In this note we review some results on the transversality conditions for a smooth Fredholm map f : X x (0, T) -> Y between two Banach spaces X, Y. These conditions are well-known in the realm of bifurcation theory and commonly accepted as "generic". Here we show that under the transversality assumptions the sections C(t) = {x : f (x, t) = 0} of the zero set of f are discrete for every t is an element of(0, T) and we discuss a somehow explicit family of perturbations of f along which transversality holds up to a residual set. The application of these results to the case when f is the X-differential of a time-dependent energy functional E : X x (0, T) -> R and C(t) is the set of the critical points of E provides the motivation and the main example of this paper.