The dimension of finite approximation dim_Fξ is defined for any point ξ of the space F(X), where F is a metrizable seminormal functor and X is a metric compactum. Such points can represent closed subsets, probability measures, maximal linked systems of closed sets, etc. The analysis shows that for many functorial constructions of general topology (exponential functor, probability measures functor, superextension functor, etc.) the dimension of finite approximation dimF ξ does not exceed the box dimension dim_B of the support supp(ξ) of this point. In this connection, the problem naturally arises of how to describe the class of functors for which the indicated restriction on the dimensionality of the finite approximation holds. This paper introduces the notion of a metrizable functor that preserves ε-nets. The condition that ε-nets be preserved by the functor F proved to be sufficient for the inequality dim_Fξ <= dim_B(supp(ξ)) for any point ξ ∈ F(X). It is proved that a number of well-known metrizable functors preserve ε-nets.