Friendship destroys orderability via clustering pseudogaps (Q1203846)

A point \(p\) is a pseudogap point of a GO space if the set of open intervals containing \(p\) is not a neighbourhood base for \(p\). A GO space \(Y\) is unbalanced if given any two sequences \(I\) and \(D\) in \(Y\) of the same confinality and no convergent proper initial segments such that \(I\) is increasing, \(D\) is decreasing, \(I<D\), and \(\{p: I< p<D\}\) is finite, then at least one of these sequences converges but \(\overline I\cap \overline D=\emptyset\). For \(p\in P\) define \(F_ p\) to be the largest convex subset containing \(p\) such that the interior of \(F_ p\) is contained in \(X- P\). Then \(p\) is said to be unfriendly (relative to \(P\)) if \(F_ p= \{p\}\) or under each admissible order on \(X\) either \(F_ p\) is not convex or has an endpoint in \(\text{int }F_ p\). Main results: (1) Let \(\overline P\) be unbalanced and each point of \(P\) be unfriendly. Then \(X\) is not orderable. (2) Let \(Y\) be a GO space, \(S\) a set of pseudogap points of \(Y\), and \(T\subset\overline S\) such that \(T\) is dense-in-itself and \(S\) is order-two-sided relative to \(T\) in \(Y\). If \(\overline S\) is unbalanced, then \(Y\) is not orderable.