LAUSR

The following general question is considered by A.V. Arhangel’skiˇi [Perfect mappings in topological groups, cross-complementary subsets and quotients, Comment. Math.Univ.Carolin., 2003]. Suppose that G is a topological group, and F,M are subspaces of G such that G = MF. Under these general assumptions, how are the properties of F andM related to the properties of G? Also, A.V. Arhangel’skiˇi and M. Tkachenko [Topological Groups and Related Structures, Atlantis Press, World Sci., 2008] asked what is about the above question in paratopological groups [Open problem 4.6.9, Topological Groups and Related Structures, Atlantis Press, World Sci., 2008]. In this paper, we mainly consider this question and some positive answers to this question are given. In particular, we find many A.V. Arhangel’skiˇi’s results hold for k-gentle paratopological groups.