## M. Tkachenko

*Homomorphic images of $\Bbb R$-factorizable groups *

Comment.Math.Univ.Carolin. 47,3 (2006) 525-537. **Abstract:**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.

**Keywords:** $\Bbb R$-factorizable, totally bounded, $\omega $-narrow, complete, Lindel\"of, $P$-space, realcompact, Dieudonn\'e-complete, pseudo-$\omega _1$-compact

**AMS Subject Classification:** Primary 54H11, 22A05, 54G10; Secondary 54D20, 54G20, 54C10, 54C45, 54D60

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