# Article

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Keywords:
$\Bbb R$-factorizable; totally bounded; $\omega$-narrow; complete; Lindelöf; $P$-space; realcompact; Dieudonné-complete; pseudo-$\omega _1$-compact
Summary:
It is well known that every $\Bbb R$-factorizable group is $\omega$-narrow, but not vice versa. One of the main problems regarding $\Bbb R$-factorizable groups is whether this class of groups is closed under taking continuous homomorphic images or, alternatively, whether every $\omega$-narrow group is a continuous homomorphic image of an $\Bbb R$-factorizable group. Here we show that the second hypothesis is definitely false. This result follows from the theorem stating that if a continuous homomorphic image of an $\Bbb R$-factorizable group is a $P$-group, then the image is also $\Bbb R$-factorizable.
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