A note on rectifiable spaces

TL;DR

This paper investigates properties of rectifiable spaces, including their homeomorphism to certain groups, local compactness, sequential properties, metrizability, and remainders, providing new insights and partial answers to longstanding questions.

Contribution

It offers new results on the structure and properties of rectifiable spaces, including conditions for local compactness, sequentiality, and metrizability, and addresses open questions in the field.

Findings

Locally compact and separable rectifiable spaces are σ-compact.

A rectifiable space is strongly Fréchet-Urysohn iff it is α₄-sequential.

Partial answers to metrizability and remainders of rectifiable spaces.

Abstract

In this paper, we firstly discuss the question: Is $l_{2}^{\infty}$ homeomorphic to a rectifiable space or a paratopological group? And then, we mainly discuss locally compact rectifiable spaces, and show that a locally compact and separable rectifiable space is $\sigma$-compact, which gives an affirmative answer to A.V. Arhangel'ski\v{i} and M.M. Choban's question [On remainders of rectifiable spaces, Topology Appl., 157(2010), 789-799]. Next, we show that a rectifiable space $X$ is strongly Fr$\acute{e}$chet-Urysohn if and only if $X$ is an $\alpha_{4}$-sequential space. Moreover, we discuss the metrizabilities of rectifiable spaces, which gives a partial answer for a question posed in \cite{LFC2009}. Finally, we consider the remainders of rectifiable spaces, which improve some results in \cite{A2005, A2007, A2009, Liu2009}.