# zbMATH — the first resource for mathematics

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)
Full Text: