When does a topological group
G have a Hausdorff compactification
bG with a remainder belonging to a given class of spaces? In this paper, we mainly improve some results of A. V. Arhangel'skiĭ and C. Liu's. Let
G be a non-locally compact topological group and
bG be a compactification of
G. The following facts are established: (1) If
bG\
G has locally a
k-space with a point-countable
k-network and π-character of
bG\
G is countable, then
G and
bG are separable and metrizable; (2) If
bG\
G has locally a δθ-base, then
G and
bG are separable and metrizable; (3) If
bG\
G has locally a quasi-
Gδ-diagonal, then
G and
bG are separable and metrizable. Finally, we give a partial answer for a question, which was posed by C. Liu in [16].
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