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

For every functional functor $F:Comp\to Comp$ in the category $Comp$ of compact Hausdorff spaces we define the notions of $F$-Dugundji and $F$-Milutin spaces, generalizing the classical notions of a 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 $F:Comp\to Comp$ admitting tensor products each Dugundji compact 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 $a\in F(\{0\})\subset F(\{0,1\})$ and $b\in F(\{1\})\subset F(\{0,1\})$ can be linked by a continuous path in $F(\{0,1\})$. We prove that for the functor $Lip_k$ of $k$-Lipschitz functionals with $k<2$, each absolute $Lip_k$-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 $Lip_3$-valued retract. More generally, each hereditarily paracompact scattered compact space $X$ of finite scattered height $n$ is an absolute $Lip_k$-valued retract for $k=2^{n+2}-1$.