Selected Ordered Space Problems

This chapter discusses selected ordered space problems. A generalized ordered space (a GO-space) is a triple (X, _T{hooktop}, <) where (X, <) is a linearly ordered set and _T{hooktop} is a Hausdorfftopology on X that has a base of order-convex sets. If _T{hooktop} is the usual open interval topology of the order <, then it is said that (X, _T{hooktop}, <) is a linearly ordered topological space (LOTS). Besides the usual real line, the most familiar examples of GO-spaces are the Sorgenfrey line, the Michael line, the Alexandroffdouble arrow, and various spaces of ordinal numbers. The chapter discusses about most important open question in GO-space theory-Maurice's problem-which Qiao and Tall showed is closely related to several other old questions of Heath and Nyikos. Some of the open problems in the theory of ordered spaces are also discussed in the chapter. For many of the questions, only definitions and references for the question are provided.