\(F\)-Dugundji spaces, \(F\)-Milutin spaces and absolute \(F\)-valued retracts (Q465846)

This article is dedicated to the discussion that the class of \(F\)-Dugundji spaces coincides with the class of absolute \(F\)-valued retracts. It is shown that for a monomorphic continuous functor \(F:\,Comp\,\to \,Comp\) admitting tensor products each Dugundji compact space is an absolute \(F\)-valued retract if and only if the doubleton \(\left\{ 0,\,1 \right\}\) is an absolute \(F\)-valued retract if and only if some points \(\alpha \in F\left( \left\{ 0 \right\} \right)\subset F\left( \left\{ 0,\,1 \right\} \right)\) and \(\beta \in F\left( \left\{ 1 \right\} \right)\subset F\left( \left\{ 0,\,1 \right\} \right)\) can be linked by a continuous path in \(F\left( \left\{ 0,\,1 \right\} \right).\) It is proved that for the functor \(Li{{p}_{k}}\) of \(k\)-Lipschitz functionals with \(k<2\), each absolute \(Li{{p}_{k}}\)-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 \(Li{{p}_{3}}\)-valued retract. More generally, each hereditarily paracompact scattered compact space \(X\) of finite scattered height \(n=ht\left( X \right)\) is an absolute \(Li{{p}_{k}}\)-valued retract for \(k={{2}^{n+2}}-1\).