| Title:
|
Cellularity and the index of narrowness in topological groups (English) |
| Author:
|
Tkachenko, M. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
52 |
| Issue:
|
2 |
| Year:
|
2011 |
| Pages:
|
309-315 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| 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 $. (English) |
| Keyword:
|
cellularity |
| Keyword:
|
$G_\delta$-modification |
| Keyword:
|
index of narrowness |
| Keyword:
|
$\omega $-narrow |
| Keyword:
|
weakly Lindelöf |
| Keyword:
|
$\mathbb R$-factorizable |
| Keyword:
|
complexity of functions |
| MSC:
|
54A25 |
| MSC:
|
54C30 |
| MSC:
|
54H11 |
| idZBL:
|
Zbl 1240.54109 |
| idMR:
|
MR2849053 |
| . |
| Date available:
|
2011-05-17T08:44:26Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/141492 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
[3] Juhász I.: Cardinal Functions in Topology.Math. Centre Tracts 34, Amsterdam, 1971. MR 0340021 |
| Reference:
|
[4] Pasynkov B.A.: On the relative cellularity of Lindelöf subspaces of topological groups.Topology Appl. 57 (1994), 249–258. Zbl 0803.54016, MR 1278026, 10.1016/0166-8641(94)90052-3 |
| Reference:
|
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| Reference:
|
[6] Tkachenko M.G.: Subgroups, quotient groups and products of $\mathbb R$-factorizable groups.Topology Proc. 16 (1991), 201–231. MR 1206464 |
| Reference:
|
[7] Tkachenko M.G.: Introduction to topological groups.Topology Appl. 86 (1998), 179–231. Zbl 0955.54013, MR 1623960, 10.1016/S0166-8641(98)00051-0 |
| Reference:
|
[8] V. V. Uspenskij, A topological group generated by a Lindelöf $\Sigma$-space has the Souslin property: Soviet Math. Dokl..26 (1982), 166–169. |
| Reference:
|
[9] Uspenskij V.V.: On continuous images of Lindelöf topological groups.Soviet Math. Dokl. 32 (1985), 802–806. Zbl 0602.22003 |
| . |
