The Priestley separation axiom for scattered spaces (Q1611299)

The authors give a new characterization of scattered compact Hausdorff spaces. Main results: (1) Let \(X\) be a scattered compact Hausdorff space with a quasi-order \(R\). Then \(R\) is closed if and only if \((X,R)\) is a Priestley space. (2) Let \(X\) be a non-scattered compact Hausdorff space. Then there is a closed equivalence relation \(E\) on \(X\) such that \((X,E)\) is not a Priestley space. (3) Let \(X\) be a non-scattered compact Hausdorff metrizable space. Then there is a closed partial order \(\leq\) on \(X\) such that \((X,\leq)\) is not a Priestley space.