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)⩽2ωw(X)⩽2ω; (2) If a nowhere locally compact space X   with locally a GδGδ-diagonal has a remainder that is a paracompact p  -space, then w(X)=ωw(X)=ω; (3) If a non-locally compact paratopological group G has a developable remainder Y  , then nw(G)=πw(G)=πw(Y)=ωnw(G)=πw(G)=πw(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)=ωw(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)=ωw(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].