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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
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