A remark on Remainders of homogeneous spaces in some compactifications

We prove that a remainder Y of a non-locally compactý ýrectifiable space X is locally a p-space if and only ifý ýeither X is a Lindel\"{o}f p-space or X is σ-compactý, ýwhich improves two results by Arhangelskiiý. ýWe also show that if a non-locally compactý ýrectifiable space X that is locally paracompact has a remainder Y which has locally a Gδ-diagonalý, ýthen both X and Y are separable and metrizableý, ýwhich improves anotherý ýArhangelskii's resultý. ýIt is proved that if a non-locally compact paratopological group G has a locally developable remainder Yý, ýthen either G and Y are separable and metrizableý, ýor G is a σ-compact space with a countable networký, ýwhich improves a result by Wang-He.