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$\Bbb R$-factorizable group; $z$-embedded set; $\aleph_0$-bounded group; $P$-group; Lindelöf group
The properties of $\Bbb R$-factorizable groups and their subgroups are studied. We show that a locally compact group $G$ is $\Bbb R$-factorizable if and only if $G$ is $\sigma$-compact. It is proved that a subgroup $H$ of an $\Bbb R$-factorizable group $G$ is $\Bbb R$-factorizable if and only if $H$ is $z$-embedded in $G$. Therefore, a subgroup of an $\Bbb R$-factorizable group need not be $\Bbb R$-factorizable, and we present a method for constructing non-$\Bbb R$-factorizable dense subgroups of a special class of $\Bbb R$-factorizable groups. Finally, we construct a closed $G_{\delta}$-subgroup of an $\Bbb R$-fac\-torizable group which is not $\Bbb R$-factorizable.
[1] Comfort W.W.: Compactness like properties for generalized weak topological sums. Pacific J. Math. 60 (1975), 31-37. MR 0431088 | Zbl 0307.54016
[2] Comfort W.W., Ross K.A.: Pseudocompactness and uniform continuity in topological groups. Pacific J. Math. 16 (1966), 483-496. MR 0207886 | Zbl 0214.28502
[3] Guran I.I.: On topological groups close to being Lindelöf. Soviet Math. Dokl. 23 (1981), 173-175. Zbl 0478.22002
[4] Hernández S., Sanchiz M., Tkačenko M.: Bounded sets in spaces and topological groups. submitted for publication.
[5] Engelking R.: General Topology. Heldermann Verlag, 1989. MR 1039321 | Zbl 0684.54001
[6] Pontryagin L.S.: Continuous Groups. Princeton Univ. Press, Princeton, 1939. Zbl 0659.22001
[7] Tkačenko M.G.: Subgroups, quotient groups and products of $\Bbb R$-factorizable groups. Topology Proceedings 16 (1991), 201-231. MR 1206464
[8] Tkačenko M.G.: Factorization theorems for topological groups and their applications. Topology Appl. 38 (1991), 21-37. MR 1093863
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