A remark on Remainders of homogeneous spaces in some compactifications

‎We prove that a remainder $Y$ of a non-locally compact‎ ‎rectifiable space $X$ is locally a $p$-space if and only if‎ ‎either $X$ is a Lindel"{o}f $p$-space or $X$ is $sigma$-compact‎, ‎which improves two results by Arhangel'skii‎. ‎We also show that if a non-locally compact‎ ‎rectifiable space $X$ that is locally paracompact has a remainder $Y$ which has locally a $G_{delta}$-diagonal‎, ‎then both $X$ and $Y$ are separable and metrizable‎, ‎which improves another‎ ‎Arhangel'skii's result‎. ‎It is proved that if a non-locally compact paratopological group $G$ has a locally developable remainder $Y$‎, ‎then either $G$ and $Y$ are separable and metrizable‎, ‎or $G$ is a $sigma$-compact space with a countable network‎, ‎which improves a result by Wang-He.