Cross-complementary subsets in paratopologicalgroups

The following general question is considered by A.V. Arhangel’skiˇı [Perfect mappings intopological groups, cross-complementary subsets and quotients, Comment. Math. Univ.Carolin. 2003]. Suppose thatGis a topological group, andF, Mare subspaces ofGsuchthatG=M F. Under these general assumptions, how are the properties ofFandMrelated to the properties ofG? Also, A.V. Arhangel’skiˇı and M. Tkachenko [TopologicalGroups and Related Structures, Atlantis Press, World Sci., 2008] asked what is about theabove question in paratopological groups [Open problem 4.6.9, Topological Groups andRelated Structures, Atlantis Press, World Sci. 2008]. In this paper, we mainly considerthis question and some positive answers to this question are given. In particular, we findmany A.V. Arhangel’skiˇı’s results hold fork-gentle paratopological groups.

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