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Keywords:
$\Bbb R$-factorizable; totally bounded; $\omega $-narrow; complete; Lindelöf; $P$-space; realcompact; Dieudonné-complete; pseudo-$\omega _1$-compact
$\Bbb R$-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.
References:
[1] Arhangel'skii A.V.: Topological Function Spaces. Math. Appl. (Soviet Series), 78, Kluwer, Dordrecht, 1992.
[2] Clay E., Clark B., Schneider V.: Using a set to construct a group topology. Kyungpook Math. J. 37 (1997), 113-116. MR 1454775 | Zbl 0876.22003
[3] Comfort W.W.: Compactness-like properties for generalized weak topological sums. Pacific J. Math. 60 (1975), 31-37. MR 0431088 | Zbl 0307.54016
[4] Guran I.: On topological groups close to being Lindelöf. Soviet Math. Dokl. 23 (1981), 173-175. Zbl 0478.22002
[5] Hernández C.: Topological groups close to being $\sigma$-compact. Topology Appl. 102 (2000), 101-111. MR 1739266
[6] Hernández C., Tkachenko M.G.: Subgroups of $\Bbb R$-factorizable groups. Comment. Math. Univ. Carolin. 39 (1998), 371-378. MR 1651979
[7] Leischner M.: Quotients of Raikov-complete topological groups. Note Mat. 13 (1993), 75-85. MR 1283519 | Zbl 0847.54030
[8] Tkachenko M.G.: Factorization theorems for topological groups and their applications. Topology Appl. 38 (1991), 21-37. MR 1093863 | Zbl 0722.54039
[9] Tkachenko M.G.: Subgroups, quotient groups and products of $\Bbb R$-factorizable groups. Topology Proc. 16 (1991), 201-231. MR 1206464
[10] Tkachenko M.G.: Introduction to topological groups. Topology Appl. 86 (1998), 179-231. MR 1623960 | Zbl 0955.54013
[11] Tkachenko M.G.: $\Bbb R$-factorizable groups and subgroups of Lindelöf $P$-groups. Topology Appl. 136 (2004), 135-167. MR 2023415 | Zbl 1039.54020
[12] Williams S.W.: Box products. in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan, eds., Chapter 4, North-Holland, Amsterdam, 1984, pp.169-200. MR 0776623 | Zbl 0769.54008
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