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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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