Two questions on rectifiable spaces and related conclusions

In the first part of this note, we answer two open questions on rectifiable spaces.We show that if C is a compact subset of a rectifiable space G and F is closed in G then C⋅F is closed in G. This answers a question of [F.C. Lin, R.X. Shen, On rectifiable spaces and paratopological groups, Topology Appl. 158 (2011) 597-610]. We also show that every rectifiable p-space with a countable Souslin number is Lindelöf. This gives a positive answer to a question of [A.V. Arhangelʼskii, M.M. Choban, Remainders of rectifiable spaces, Topology Appl. 157 (2010) 789-799].In the last part of this note we point out that a non-locally compact rectifiable paracompact space has the following conclusion:If a non-locally compact rectifiable paracompact space G has a compactification bG such that the remainder bG∖G of G belongs to P, then G and bG∖G are separable and metrizable, where P is a class of spaces which satisfies the following conditions:(1)if X∈P, then every compact subset of the space X is a Gδ-set of X;(2)if X∈P and X is not locally compact, then X is not locally countably compact;(3)if X∈P and X is a Lindelöf p-space, then X is metrizable.Some known conclusions on rectifiable paracompact spaces and their remainders can be gotten by this conclusion.