A note on rectifiable spaces

In this paper, we firstly discuss the question: Is l 2 ∞ homeomorphic to a rectifiable space or a paratopological group? And then, we mainly discuss locally compact rectifiable spaces, and show that a locally compact rectifiable space with the Souslin property is σ-compact, which gives an affirmative answer to A.V. Arhangelʼskiı̌ and M.M. Chobanʼs question [A.V. Arhangelʼskiı̌, M.M. Choban, On remainders of rectifiable spaces, Topology Appl. 157 (2010) 789–799]. Next, we show that a rectifiable space X is strongly Fréchet–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 F.C. Lin and R.X. Shen (2011) [16]. Finally, we consider the remainders of rectifiable spaces, which improve some results in A.V. Arhangelʼskiı̌ (2005) [2], A.V. Arhangelʼskiı̌ and M.M. Choban (2010) [5], C. Liu (2009) [17].