Remainders and cardinal invariants

In this paper, we investigate remainders and cardinal invariants of some topological spaces (or semitopological groups, paratopological groups). The main results are: (1) If a non-locally compact homogeneous space X is locally ccc and X has a remainder with a locally point-countable base, then w ( X ) ⩽ 2 ω ; (2) If a nowhere locally compact space X with locally a G δ -diagonal has a remainder that is a paracompact p-space, then w ( X ) = ω ; (3) If a non-locally compact paratopological group G has a developable remainder Y, then n w ( G ) = π w ( G ) = π w ( Y ) = ω ; (4) If a non-locally compact paratopological group G has a remainder Y with a point-countable base, then w ( G ) = w ( Y ) = ω ; (5) If a semitopological group H is r-equivalent to a non-locally compact semitopological group G that has a countable base, then w ( H ) = ω . Among them, (2) generalizes a result by A.V. Arhangelʼskii [1, Theorem 4.2], (4) generalizes both A.V. Arhangelʼskiiʼs result [5, Theorem 10] and C. Liuʼs result [14, Theorem 3.1], and (5) generalizes a result by A.V. Arhangelʼskii [2, Theorem 4.7].