On metric order in spaces of the form F(X)

For a metrizable functor F and a point ξ∈F(Y) (Y is a compact metric space) we define lower and upper metric orders o_(ξ) and o‾(ξ) as a numerical characteristic of an approximation of ξ by points ξn∈Fn(Y). If F is the exponential functor exp then o_(ξ) and o‾(ξ) coincide, respectively, with classical lower and upper capacitarian dimensions dim_Bξ and dim‾Bξ of a closed subset ξ⊂Y. We establish some properties of o_(ξ) and o‾(ξ) and pose several questions.