This paper deals with smooth optimization problems P in R(n) depending on parameter y is an element of R(p). The problem P(y) is defined by means of a finite number of equality and inequality constraints. We study the set Sigma(KKT) of pairs (x, y) such that x is a Karush-Kuhn-Tucker point of the problem P(y). Let Sigma denote the subset of Sigma(KKT) at which the Mangasarian-Fromovitz constraint qualification is fulfilled. For problem data in general position we prove that Sigma is a topological manifold of dimension p.