F-7 - Linearly Ordered and Generalized Ordered Spaces

Abstract

For any linearly ordered set (X, <), if Ι (<) is the topology on X that has the collection of all open intervals of (X, <) as a base, the topology Ι (<) is the open interval topology (or order topology) of the order < and (X, <, Ι (<)) is a linearly ordered topological space (LOTS). For a subset Y ⊆X, it can happen that the relative topology Ι (<)|Y on Y does not coincide with the open interval topology Ι (<|Y) induced on Y by the restricted ordering. In some cases, there might be some other ordering of Y, whose open interval topology coincides with Ι (<)|Y. The best known example is the Sorgenfrey line. The study of the subspaces of LOTS in their own right was pioneered by E. Čech who introduced generalized ordered spaces (GO-spaces), that is, triples (X,<, ▪) where < is a linear ordering of the set X and ▪is a Hausdorff topology on X such that each point of X has a ▪-neighborhood base consisting of (possibly degenerate) intervals. Such spaces have also been called suborderable spaces.