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...
