Cross-complementary subsets in paratopological groups

The following general question is considered by A.V. Arhangel’skii [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 $F$ and $M$ related to the properties of $G$? Also, A.V. Arhangel’skii 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’skii's results hold for $k$-gentle paratopological groups.

Keywords:

metrizable groups, paratopological groups, $k$-gentle paratopological groups perfect mappings, paracompact $p$-spaces, countable tightnesses,

PDF

___

[1] A.V. Arhangel’skii, Perfect mappings in topological groups, cross-complementary sub- sets and quotients, Comment. Math. Univ. Carolin. 44, 4, 701–709, 2003.
[2] A.V. Arhangel’skii and V.I. Ponomarev, Fundamentals of General Topology: Problems and Exercises, Reidel, 1984.
[3] A.V. Arhangel’skii and M. Tkachenko, Topological Groups and Related Structures, Atlantis Press, World Sci. 2008.
[4] N. Bourbaki, Elements de Mathematique, Premiere Partie, Livre 3, Ch. 3, 3-me ed., Hermann, Paris, 1949.
[5] R. Engelking, General Topology, Revised and completed edition, Heldermann Verlag, 1989.
[6] V.V. Filippov, On perfect images of paracompact p-spaces, Soviet Math. Dokl. 176, 533–536, 1967.
[7] P. Li and L. Mou, On quasitopological groups, Topology Appl. 161, 243–247, 2014.
[8] O. Ravsky, On H-closed paratopological groups, Visnyk Lviv. Univ. Ser. Mat. Mekh. 59, 96–101, 2001.
[9] O. Ravsky, The topological and algebraic properties of paratopological groups, Ph.D. thesis. Lviv University, 2003 (in Ukrainian).
[10] W. Roelke and S. Dierolf, Uniform Structures on Topological Groups and Their Quo- tients, McGraw-Hill, New York, 1981.
[11] M. Tkachenko, Paratopological and semitopological groups vs topological groups, in: K.P. Hart, J. van Mill, P. Simon (Eds.), Recent Progress in General Topology III, Springer Atlantis Press, 803–859, 2013.