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On continuous images of Lindelöf topological groups. (English. Russian original) Zbl 0602.22003
Sov. Math., Dokl. 32, 802-806 (1985); translation from Dokl. Akad. SSSR 285, 824-827 (1985).
Topological groups generated by a Lindelöf $$\Sigma$$-space (in the sense of Nagami) and their continuous images are under investigation. It is shown that if a Lindelöf $$\Sigma$$-space generates a topological group G then any family of $$G_{\delta}$$-subsets of G contains a countable dense subfamily. The main result of the paper is the following one. Let a topological group G be generated by its Lindelöf $$\Sigma$$-space, X dense in G and $$F: X\to Y$$ a continuous mapping onto a regular space Y. Then the net weight of the set $$\{$$ $$y\in Y:\chi$$ (y,Y)$$\leq \tau \}$$ does not exceed $$\tau$$ for each cardinal number $$\tau\geq \aleph 0$$ (Theorem 7). Moreover, if Y is compact then $$\chi (Y)=w(Y)$$ and any infinite regular cardinal number is a caliber for Y.
A note. The reviewer has now proved that $$w(Y)=t(Y)$$ for any compact space Y which is a continuous image of a topological group generated by a Lindelöf $$\Sigma$$-space. This answers the question of Arkhangel’skij formulated in the present paper.
Reviewer: M.G.Tkachenko

##### MSC:
 22A05 Structure of general topological groups 54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets) 54C05 Continuous maps