Abstract:We study relations between the cellularity and index of narrowness in topological groups and their $G_\delta$-modifications. We show, in particular, that the inequalities $\operatorname{in} ((H)_\tau)\le 2^{\tau\cdot \operatorname{in} (H)}$ and $c((H)_\tau)\leq 2^{2^{\tau\cdot \operatorname{in} (H)}}$ hold for every topological group $H$ and every cardinal $\tau\geq \omega $, where $(H)_\tau$ denotes the underlying group $H$ endowed with the $G_\tau$-modification of the original topology of $H$ and $\operatorname{in} (H)$ is the index of narrowness of the group $H$. Also, we find some bounds for the {\it complexity\/} of continuous real-valued functions $f$ on an arbitrary $\omega $-narrow group $G$ understood as the minimum cardinal $\tau\geq \omega $ such that there exists a continuous homomorphism $\pi\colon G\to H$ onto a topological group $H$ with $w(H)\leq \tau$ such that $\pi\prec f$. It is shown that this complexity is not greater than $2^{2^\omega }$ and, if $G$ is weakly Lindel\"of (or {\it $2^\omega $-steady\/}), then it does not exceed $2^\omega $.
Keywords: cellularity, $G_\delta$-modification, index of narrowness, $\omega $-narrow, weakly Lindel\"of, $\mathbb R$-factorizable, complexity of functions
AMS Subject Classification: 54H11 54A25 54C30