# Cellularity and the index of narrowness in topological groups

Commentationes Mathematicae Universitatis Carolinae (2011)

- Volume: 52, Issue: 2, page 309-315
- ISSN: 0010-2628

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topTkachenko, 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 -

## References

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