  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}.
