This paper is devoted to the investigation of cardinal invariants such as the local density, the local weak density and the relation between the tightness of the space C n (X) of closed sets with finitely many components and the density of a topological space itself. Moreover, it is shown that the functor C n : Comp → Comp preserves the local density and the local weak density of compact spaces. As a result, criteria for locally separability and locally weakly separability of compact spaces are obtained.