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Cellularity and the index of narrowness in topological groups. (English) Zbl 1240.54109
I. Juhász [Cardinality functions in topology, Mathematical Centre Tracts 34 (1971; Zbl 0224.54004)] proved that \(c(X)_{\kappa }\leq 2^{\kappa c(X)}\), where \(X\) is a compact Hausdorff space, \(c(X)\) is its cellularity, \(\kappa \) is an infinite cardinal and \((X)_{\kappa }\) has \(G_{\kappa }\)-sets for its open base. In the present paper, a similar result is proved for topological groups \(G\: c(G)_{\kappa }\leq 2^{2^{\kappa \text{in}(G)}}\), where \(\text{in}(G)\) is the index of narrowness, i.e., \(\min \{\kappa \geq \omega; G\) can be covered by \(\kappa \) shifts of every neighborhood of identity\(\}\). In particular, \(c(G)_{\omega }\leq 2^{2^{c(G)}}\) and that upper limit can be reached (the constructed example is a subspace of a power of an Abelian free group).
The second part deals with a possible generalization to \(\omega \)-narrow groups \(G\) of a characterizing property of \(\mathbb {R}\)-factorizable groups: a factorization of continuous real-valued functions on \(G\) via continuous homomorphisms onto groups \(K\) with small \(w(K)\). It is shown that one can always require \(w(K)\leq 2^{2^{\omega }}\) and that the upper limit can be decreased to \(2^{\omega }\) if \(G\) is weakly Lindelöf (then the factorized map may have a more general range, namely any Tikhonov space \(X\) with \(w(X)\leq 2^{\omega }\)) or when \(G\) is \(2^{\omega }\)-steady.
Three interesting open problems are posed at the end of the paper.
MSC:
54H11 Topological groups (topological aspects)
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
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