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Transversal group topologies on non-Abelian groups. (English) Zbl 1104.22001
Two topologies on a space \(X\) are called transversal if their union generates the discrete topology on \(X\) [see M. G. Tkačenko, V. V. Tkachuk, R. G. Wilson and I. V. Yaschenko, Proc. Am. Math. Soc. 128, 287–297 (2000; Zbl 0932.54035)]. The authors continue their study of transversal group topologies that they began in an earlier paper and obtain several interesting results. For example, the linear matrix groups \(GL_n(\mathbb R)\) and \(GL_n(\mathbb C)\) admit transversal topologies, but \(SL_n(\mathbb R)\) and \(SL_n(\mathbb C)\) do not admit such topologies. If a locally compact group admits a transversal topology, then it is neither connected nor separable. A locally compact connected group admits a transversal group topology if and only if its center is either transversable or discrete; in particular, a connected Lie group admits a transversal topology if and only if its center is not compact. There is a set \(X\) admitting transversal Tychonoff topologies \(\tau_1\) and \(\tau_2\), such that the free group topologies of \(F(X,\tau_1)\) and \(F(X,\tau_2)\) are not transversal. Four open questions are posed.

MSC:
22A05 Structure of general topological groups
54H11 Topological groups (topological aspects)
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54G20 Counterexamples in general topology
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[1] Anderson, B.A., A class of topologies with \(T_1\)-complements, Fund. math., 69, 267-277, (1970) · Zbl 0203.55401
[2] Anderson, B.A., Families of mutually complementary topologies, Proc. amer. math. soc., 29, 362-368, (1971) · Zbl 0214.49404
[3] Bagley, R., On the characterization of the lattice of topologies, J. London math. soc., 30, 247-249, (1955) · Zbl 0058.16505
[4] Birkhoff, G., On the combination of topologies, Fund. math., 26, 156-166, (1936) · Zbl 0014.28002
[5] Dikranjan, D., Recent advances in minimal topological groups, Topology appl., 126, 149-168, (1998)
[6] Dikranjan, D.; Prodanov, I.; Stoyanov, L., Topological groups: characters, dualities and minimal group topologies, Pure and applied mathematics, vol. 130, (1989), Marcel Dekker New York
[7] Dikranjan, D.; Tkachenko, M.; Yaschenko, I., On transversal group topologies I, Topology appl., 153, 786-817, (2005) · Zbl 1114.22001
[8] Engelking, R., General topology, (1989), Heldermann Berlin · Zbl 0684.54001
[9] Gaughan, E.D., Topological group structures of infinite symmetric groups, Proc. nat. acad. sci. USA, 58, 907-910, (1967) · Zbl 0153.04301
[10] Graev, M.I., Free topological groups, (), Izv. akad. nauk SSSR ser. mat., 12, 279-323, (1948), Russian original in:
[11] Guran, I., On topological groups close to being Lindelöf, Soviet math. dokl., 23, 173-175, (1981) · Zbl 0478.22002
[12] Hewitt, E.; Ross, K., Abstract harmonic analysis, vol. I, (1979), Springer Berlin
[13] Juhász, I.; Tkachenko, M.; Tkachuk, V.; Wilson, R., Self-transversal spaces and their discrete subspaces, Rocky mountain J. math., 35, 4, 1157-1172, (2005) · Zbl 1142.54017
[14] Megrelishvili, M., Group representations and construction of minimal topological groups, Topology appl., 62, 1, 1-19, (1995) · Zbl 0824.22003
[15] Prodanov, I., Maximal and minimal topologies on abelian groups, (), 985-997 · Zbl 0452.22002
[16] Prodanov, I.R.; Stoyanov, L.N., Every minimal abelian group is precompact, Dokl. bulg. acad. sci., 37, 23-26, (1984) · Zbl 0546.22001
[17] Remus, D.; Stoyanov, L., Complete minimal and totally minimal groups, Topology appl., 42, 57-69, (1991) · Zbl 0758.22001
[18] van Rooij, A., The lattice of all topologies is complemented, Canad. J. math., 20, 805-807, (1968) · Zbl 0157.53003
[19] Shakhmatov, D.; Tkachenko, M.; Wilson, R., Transversal and \(T_1\)-independent topologies, Houston J. math., 30, 2, 421-433, (2004) · Zbl 1061.54002
[20] Steiner, A., The topological complementation problem, Bull. amer. math. soc., 72, 125-127, (1966) · Zbl 0134.18206
[21] Steiner, A., Complementation in the lattice of \(T_1\)-topologies, Proc. amer. math. soc., 17, 884-886, (1966) · Zbl 0139.40201
[22] Steiner, A., The lattice of topologies: structure and complementation, Trans. amer. math. soc., 122, 379-398, (1966) · Zbl 0139.15905
[23] Steiner, E.; Steiner, A., Topologies with \(T_1\)-complements, Fund. math., 61, 23-28, (1967) · Zbl 0162.26202
[24] Steiner, A.; Steiner, E., A \(T_1\)-complement for the reals, Proc. amer. math. soc., 19, 177-179, (1968) · Zbl 0162.26301
[25] Stephenson, R.M., Minimal topological groups, Math. ann., 192, 193-195, (1971) · Zbl 0206.31601
[26] Tkachenko, M.G., Introduction to topological groups, Topology appl., 86, 179-231, (1998)
[27] Tkačenko, M.G.; Tkachuk, V.V.; Wilson, R.G.; Yaschenko, I., No submaximal topology on a countable set is \(T_1\)-complementary, Proc. amer. math. soc., 128, 1, 287-297, (2000) · Zbl 0932.54035
[28] Torres Falcón, Y., Union of chains of subgroups of a topological group, Appl. gen. topology, 2, 2, 227-235, (2001) · Zbl 1018.54004
[29] Zelenyuk, E.; Protasov, I., Complemented topologies on abelian groups, Sibirsk. mat. zh., Siberian math. J., 42, 3, 465-472, (2001), ii (in Russian); English transl. in: · Zbl 0997.22002
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