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Keywords:
cellularity; $G_\delta$-modification; index of narrowness; $\omega $-narrow; weakly Lindelöf; $\mathbb R$-factorizable; complexity of functions
cellularity; $G_\delta$-modification; index of narrowness; $\omega $-narrow; weakly Lindelöf; $\mathbb R$-factorizable; complexity of functions
Summary:
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 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 Lindelöf (or $2^\omega $-steady), then it does not exceed $2^\omega $.
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