Topologically subordered rectifiable spaces and compactifications

Abstract

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\phi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \phi =\pi_{1}$ and for every $x\in G$ we have $\phi (x, x)=(x, e)$, where $\pi_{1}: G\times G\rightarrow G$ 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. Arhangel'ski\v\i\ in \cite{A20092}. As an applications, 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 $\Phi$ it is true. Statement: Suppose that $G$ is a non-locally compact GO-space which is rectifiable, and that $Y=bG\setminus G$ has (locally) a property-$\Phi$. Then $G$ and $bG$ are separable and metrizable. Moreover, we also consieder some related matters about the remainders of the Hausdorff compactifications of rectifiable spaces.