Topologically subordered rectifiable spaces and compactifications

A topological space G is said to be a rectifiable space provided that there are a surjective homeomorphism φ:G×G→G×G and an element e∈G such that π1∘φ=π1 and for every x∈G we have φ(x,x)=(x,e), where π1:G×G→G is the projection to the first coordinate. In this paper, we mainly discuss the rectifiable spaces which are suborderable, and show that if a rectifiable space is suborderable then it is metrizable or a totally disconnected P-space, which improves a theorem of A.V. Arhangelʼskiı̌ (2009) in [8]. As an application, we discuss the remainders of the Hausdorff compactifications of GO-spaces which are rectifiable, and we mainly concerned with the following statement, and under what condition Φ it is true.Statement – Suppose that G is a non-locally compact GO-space which is rectifiable, and that Y=bG∖G has (locally) a property-Φ. Then G and bG are separable and metrizable.Moreover, we also consider some related matters about the remainders of the Hausdorff compactifications of rectifiable spaces.