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Keywords:
$\Bbb R$-factorizable group; $z$-embedded set; $\aleph_0$-bounded group; $P$-group; Lindelöf group
$\Bbb R$-factorizable group; $z$-embedded set; $\aleph_0$-bounded group; $P$-group; Lindelöf group
Summary:
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.
References:
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