Local properties on the remainders of the topological groups

Abstract

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$\backslash$G has locally a k-space with a point-countable k-network and π-character of bG$\backslash$G is countable, then G and bG are separable and metrizable; (2) If bG$\backslash$G has locally a δθ-base, then G and bG are separable and metrizable; (3) If bG$\backslash$G has locally a quasi-G_δ-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].