F-Dugundji spaces, F-Milutin spaces and absolute F-valued retracts

For every functional functor F:Comp→Comp in the category Comp of compact Hausdorff spaces we define the notions of F-Dugundji and F-Milutin spaces, generalizing the classical notions of Dugundji and Milutin spaces. We prove that the class of F-Dugundji spaces coincides with the class of absolute F-valued retracts. Next, we show that for a monomorphic continuous functor F:Comp→Comp admitting tensor products each Dugundji compact space is an absolute F-valued retract if and only if the doubleton {0,1} is an absolute F-valued retract if and only if some points a∈F({0})⊂F({0,1}) and b∈F({1})⊂F({0,1}) can be linked by a continuous path in F({0,1}). We prove that for the functor Lipk of k-Lipschitz functionals with k<2, each absolute Lipk-valued retract is openly generated. On the other hand, the one-point compactification of any uncountable discrete space is not openly generated but is an absolute Lip3-valued retract. More generally, each hereditarily paracompact scattered compact space X of finite scattered height n=ht(X) is an absolute Lipk-valued retract for k=2n+2−1.