A characterization of Dugundji spaces via set-valued maps

We characterize Lindelöf p-spaces which are absolute extensors for zero-dimensional perfectly normal spaces. As an application we prove that a Lindelöf Čech-complete space X is an absolute extensor for zero-dimensional spaces if and only if there exists an upper semi-continuous compact-valued map r : X3 → X such that r(x, y, y) = r(y, y, y) = x for all , y ε X. This result is new even when applied to compact spaces and yields the following new characterization of Dugundji spaces: A compact Hausdorff space X is Dugundji if and only if there exists an upper semi-continuous compact-valued map r : X3 → X such that r(x, y, y) = r(y, y, x) = x for all x, y ε X. It is worth noting that, by a result of Uspenskij, in the above characterization of Dugundji spaces the set-valued map r cannot be replaced by a single-valued (continuous) map, the 5-dimensional sphere S5 being a counterexample.