Topologically subordered rectifiable spaces and compactifications

Abstract

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 π 1 ∘ φ = π 1 and for every x ∈ G we have φ ( x , x ) = ( x , e ) , where π 1 : 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 = b G ∖ 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.