Separable zero-dimensional spaces which are continuous images of ordered compacta (Q2705777)

The authors prove a structure theorem about separable zero-dimensional spaces which are continuous images of ordered compacta. A compact, connected space \(Z\) is a dendron if for every two distinct points \(x\) and \(x'\) of \(Z\) there exists \(z\in Z\) such that \(x\) and \(x'\) belong to distinct components of \(Z-\{z\}\). A dendrite is a metrizable dendron. A non-empty, closed subset \(A\) of a dendron \(Z\) is a strong \(T\)-set in \(Z\) if each component of \(Z-A\) is homeomorphic to the real line. Let \(A\) be a strong \(T\)-set in \(Z\) and let \(C\) be a subset of \(A\) such that \(Z-\{x\}\) has exactly two components, \(K_{x,0}\) and \(K_{x,1}\), for each \(x\in C\). The dendritic resolution of \(A\) by means of \(C\) with respect to \(Z\) is the set \({\mathbf s}(A,C,Z)=(A-C)\cup(C\times\{0,1\})\) with a subbasis of open sets consisting of all sets \(\pi^{-1}(u)\) where \(U\) is open in \(A\) and of all sets \(\pi^{-1} (K_{x, i}) \cup\{(x,i)\}\) where \((x,i)\in C\times \{0,1\}\) and where \(\pi: {\mathbf s} (A,C,Z)\to A\) is defined \(\pi(s)=s\) for \(s\in A-C\) and \(\pi(x,i)=x\) for \((x,i) \in C\times \{0,1\}\). The main result of the paper is theNEWLINENEWLINENEWLINETheorem. Let \(X\) be a zero-dimensional space which is a continuous image of a separable and orderable compactum. Then there exist a dendrite \(Z\) and a strong \(T\)-set \(A\) in \(Z\) such that \(X\) is homeomorphic to a dendritic resolution of \(A\) with respect to \(Z\).NEWLINENEWLINENEWLINEThe authors also give an example of a compact, separable, zero-dimensional, monotonically normal space which is the continuous image of an ordered compactum, but is not orderable.