Geodesic
Homomorphic images of $\Bbb R$-factorizable groups
Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 3, pp. 525-537

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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.
Classification : 22A05, 54C10, 54C45, 54D20, 54D60, 54G10, 54G20, 54H11
Keywords: $\Bbb R$-factorizable; totally bounded; $\omega $-narrow; complete; Lindelöf; $P$-space; realcompact; Dieudonné-complete; pseudo-$\omega _1$-compact 
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     title = {Homomorphic images of $\Bbb R$-factorizable groups},
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Tkachenko, M. Homomorphic images of $\Bbb R$-factorizable groups. Commentationes Mathematicae Universitatis Carolinae, Tome 47 (2006) no. 3, pp. 525-537. http://geodesic.mathdoc.fr/item/CMUC_2006__47_3_a14/