Locally $\sigma$-compact rectifiable spaces

TL;DR

This paper investigates the properties of locally σ-compact rectifiable spaces, proving their paracompactness, classifying spaces with a bc-base, and establishing rectifiable completeness for k_omega spaces.

Contribution

It provides new results on the structure and properties of locally σ-compact rectifiable spaces, answering open questions and improving existing theorems.

Findings

Locally compact rectifiable spaces are paracompact.

Spaces with a bc-base are either locally compact or zero-dimensional.

k_omega-rectifiable spaces are rectifiable complete.

Abstract

A topological space $G$ is said to be a {\it rectifiable space} provided that there are a surjective homeomorphism $\varphi :G\times G\rightarrow G\times G$ and an element $e\in G$ such that $\pi_{1}\circ \varphi =\pi_{1}$ and for every $x\in G$, $\varphi (x, x)=(x, e)$, where $\pi_{1}: G\times G\rightarrow G$ is the projection to the first coordinate. In this paper, we first prove that each locally compact rectifiable space is paracompact, which gives an affirmative answer to Arhangel'skii and Choban's question (Arhangel'skii and Choban [3]). Then we prove that every locally $\sigma$-compact rectifiable space with a $bc$-base is locally compact or zero-dimensional, which improves Arhangel'skii and van Mill's result (Arhangel'skii and van Mill [4]). Finally, we prove that each $k_{\omega}$-rectifiable space is rectifiable complete.