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 Lindelof p-space or X is σ-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 Gδ-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 σ-compact space with a countable network‎, ‎which improves a result by Wang-He.