A note on rectifiable spaces

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 $σ$-compact, which gives an affirmative answer to A.V. Arhangel'skiǐ 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 $α_{4}$-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}.

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