Is every GO-topology a join of two orderable topologies?

As a generalization of GO-topologies (GO = generalized ordered), we are interested in those topologies (here called JO-topologies) on a set A1 which can be expressed as a join of orderable topologies (the join being taken in the lattice of all topologies on X). If a topology t is the join of m orderable topologies we say t is JO(m). It is not difficult to prove that every GO-topology is a JO-topology, but the question (raised in (M 71)) as to whether every GO-topology is JO(2) seems much more difficult). We show that X is JO{2) if X is a subspace of an orderable space D, where D is either metrizable and locally separable, or connected with countable cellularity. (The theorem is actually more general than is stated here.) We give an example to show that for any positive integer n there is a finite join of order topologies which is not JO(/i), but these are not GO-topologies. 1. Background and preliminary results. The notion of generalized ordered space was introduced by Cech in the 1930s (see (C, p. 286)). Since then they have been studied from various points of view by many authors (see for example (L 71, F, P 77, P 81); see Lutzer (L 80) for a recent survey with further references). Such spaces have also been called suborderable (P 77); we prefer the latter term as being more descriptive (GO-spaces are those spaces which can arise as subspaces of orderable spaces), but it does not seem to be widely used. A study of the properties of cardinal functions for the class of finite joins of orderable topologies can be found in (WM). Let (X, t, ,<) be linearly ordered topological space (LOTS) containing A- as a subspace, with t = relative topology on X. A point x in X is called a bad point of X if the /-neighborhoods of X are not the same as the induced order neighborhoods; let B denote the set of all bad points. We associate with each bad point in X one or two points in D — X (called missing points), chosen as follows: if p is a bad point from below, thenp is the A'-sup of {y G X: y <p} andp is not the D-sup of this set; i.e., there is a point m in D - X such that {y G X: y <p) <m <p. The construction for points which are bad from above is analogous. For each bad point choose a missing point below and/or above, and let M denote the set of missing points so chosen.