<P> The link between critical points and topology already appears at a lower level of abstraction . For example, let V (\ displaystyle V) be a sub-manifold of R n, (\ displaystyle \ mathbb (R) ^ (n),) and P be a point outside V . (\ displaystyle V .) The square of the distance to P of a point of V (\ displaystyle V) is a differential map such that each connected component of V (\ displaystyle V) contains at least a critical point, where the distance is minimal . It follows that the number of connected components of V (\ displaystyle V) is bounded above by the number of critical points . </P> <P> In the case of real algebraic varieties, this observation associated with Bézout's theorem allows us to bound the number of connected components by a function of the degrees of the polynomials that define the variety . </P>

What is critical points in maxima and minima