On \(K_ 0\)-functions and regular extension operators (Q811590)

The word space will refer to compact Hausdorff spaces. For any space \(X\) let \(\tau X\) denote the topology of \(X\). We call a function \(e: \tau S\to \tau X\), \(S\subset X\), a \(K_ 0\)-function if: \(e(\emptyset)=\emptyset\), \(S\cap e(U)=U\) for all \(U\in \tau S\) and \(e(U)\cap e(V)=e(U\cap V)\) for all \(U,V\in\tau S\). At several times, the existence of \(K_ 0\)-functions and regular extension operators is closely related. For example, Širokov proved that a space \(S\) is a Dugundji space if and only if there is a \(K_ 0\)-function from \(\tau S\) into \(\tau X\) for every embedding of \(S\) into an arbitrary space \(X\). It seems natural to ask whether the existence of a \(K_ 0\)-function from \(\tau S\) into \(\tau X\) is equivalent to the existence of a regular extension operator from \(C(S)\) into \(C(X)\). In this paper we solve this question by showing that neither of these two properties implies the other.