Cellularity and the index of narrowness in topological groups

Tkachenko, Mihail G.. "Cellularity and the index of narrowness in topological groups." Commentationes Mathematicae Universitatis Carolinae 52.2 (2011): 309-315. <http://eudml.org/doc/246874>.

@article{Tkachenko2011,
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 $c((H)_\tau )\le 2^\{2^\{\tau \cdot \operatorname\{in\} (H)\}\}$ hold for every topological group $H$ and every cardinal $\tau \ge \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 \ge \omega $ such that there exists a continuous homomorphism $\pi \colon G\rightarrow H$ onto a topological group $H$ with $w(H)\le \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 $.},
author = {Tkachenko, Mihail G.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {cellularity; $G_\delta $-modification; index of narrowness; $\omega $-narrow; weakly Lindelöf; $\mathbb \{R\}$-factorizable; complexity of functions; topological group; cellularity; narrow group; -topology; factorisation},
language = {eng},
number = {2},
pages = {309-315},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Cellularity and the index of narrowness in topological groups},
url = {http://eudml.org/doc/246874},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Tkachenko, Mihail G.
TI - Cellularity and the index of narrowness in topological groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 2
SP - 309
EP - 315
AB - 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 $c((H)_\tau )\le 2^{2^{\tau \cdot \operatorname{in} (H)}}$ hold for every topological group $H$ and every cardinal $\tau \ge \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 \ge \omega $ such that there exists a continuous homomorphism $\pi \colon G\rightarrow H$ onto a topological group $H$ with $w(H)\le \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 $.
LA - eng
KW - cellularity; $G_\delta $-modification; index of narrowness; $\omega $-narrow; weakly Lindelöf; $\mathbb {R}$-factorizable; complexity of functions; topological group; cellularity; narrow group; -topology; factorisation
UR - http://eudml.org/doc/246874
ER -