Homomorphic images of ℝ -factorizable groups

Mihail G. Tkachenko

Commentationes Mathematicae Universitatis Carolinae (2006)

  • Volume: 47, Issue: 3, page 525-537
  • ISSN: 0010-2628

Abstract

top
It is well known that every ℝ -factorizable group is ω -narrow, but not vice versa. One of the main problems regarding ℝ -factorizable groups is whether this class of groups is closed under taking continuous homomorphic images or, alternatively, whether every ω -narrow group is a continuous homomorphic image of an ℝ -factorizable group. Here we show that the second hypothesis is definitely false. This result follows from the theorem stating that if a continuous homomorphic image of an ℝ -factorizable group is a P -group, then the image is also ℝ -factorizable.

How to cite

top

Tkachenko, Mihail G.. "Homomorphic images of $\mathbb {R}$-factorizable groups." Commentationes Mathematicae Universitatis Carolinae 47.3 (2006): 525-537. <http://eudml.org/doc/249869>.

@article{Tkachenko2006,
abstract = {It is well known that every $\mathbb \{R\}$-factorizable group is $\omega $-narrow, but not vice versa. One of the main problems regarding $\mathbb \{R\}$-factorizable groups is whether this class of groups is closed under taking continuous homomorphic images or, alternatively, whether every $\omega $-narrow group is a continuous homomorphic image of an $\mathbb \{R\}$-factorizable group. Here we show that the second hypothesis is definitely false. This result follows from the theorem stating that if a continuous homomorphic image of an $\mathbb \{R\}$-factorizable group is a $P$-group, then the image is also $\mathbb \{R\}$-factorizable.},
author = {Tkachenko, Mihail G.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {$\mathbb \{R\}$-factorizable; totally bounded; $\omega $-narrow; complete; Lindelöf; $P$-space; realcompact; Dieudonné-complete; pseudo-$\omega _1$-compact; totally bounded; -narrow; complete; Lindelöf; -space},
language = {eng},
number = {3},
pages = {525-537},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Homomorphic images of $\mathbb \{R\}$-factorizable groups},
url = {http://eudml.org/doc/249869},
volume = {47},
year = {2006},
}

TY - JOUR
AU - Tkachenko, Mihail G.
TI - Homomorphic images of $\mathbb {R}$-factorizable groups
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2006
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 47
IS - 3
SP - 525
EP - 537
AB - It is well known that every $\mathbb {R}$-factorizable group is $\omega $-narrow, but not vice versa. One of the main problems regarding $\mathbb {R}$-factorizable groups is whether this class of groups is closed under taking continuous homomorphic images or, alternatively, whether every $\omega $-narrow group is a continuous homomorphic image of an $\mathbb {R}$-factorizable group. Here we show that the second hypothesis is definitely false. This result follows from the theorem stating that if a continuous homomorphic image of an $\mathbb {R}$-factorizable group is a $P$-group, then the image is also $\mathbb {R}$-factorizable.
LA - eng
KW - $\mathbb {R}$-factorizable; totally bounded; $\omega $-narrow; complete; Lindelöf; $P$-space; realcompact; Dieudonné-complete; pseudo-$\omega _1$-compact; totally bounded; -narrow; complete; Lindelöf; -space
UR - http://eudml.org/doc/249869
ER -

References

top
  1. Arhangel'skii A.V., Topological Function Spaces, Math. Appl. (Soviet Series), 78, Kluwer, Dordrecht, 1992. 
  2. Clay E., Clark B., Schneider V., Using a set to construct a group topology, Kyungpook Math. J. 37 (1997), 113-116. (1997) Zbl0876.22003MR1454775
  3. Comfort W.W., Compactness-like properties for generalized weak topological sums, Pacific J. Math. 60 (1975), 31-37. (1975) Zbl0307.54016MR0431088
  4. Guran I., On topological groups close to being Lindelöf, Soviet Math. Dokl. 23 (1981), 173-175. (1981) Zbl0478.22002
  5. Hernández C., Topological groups close to being σ -compact, Topology Appl. 102 (2000), 101-111. (2000) MR1739266
  6. Hernández C., Tkachenko M.G., Subgroups of ℝ -factorizable groups, Comment. Math. Univ. Carolin. 39 (1998), 371-378. (1998) MR1651979
  7. Leischner M., Quotients of Raikov-complete topological groups, Note Mat. 13 (1993), 75-85. (1993) Zbl0847.54030MR1283519
  8. Tkachenko M.G., Factorization theorems for topological groups and their applications, Topology Appl. 38 (1991), 21-37. (1991) Zbl0722.54039MR1093863
  9. Tkachenko M.G., Subgroups, quotient groups and products of ℝ -factorizable groups, Topology Proc. 16 (1991), 201-231. (1991) MR1206464
  10. Tkachenko M.G., Introduction to topological groups, Topology Appl. 86 (1998), 179-231. (1998) Zbl0955.54013MR1623960
  11. Tkachenko M.G., ℝ -factorizable groups and subgroups of Lindelöf P -groups, Topology Appl. 136 (2004), 135-167. (2004) Zbl1039.54020MR2023415
  12. Williams S.W., Box products, in: Handbook of Set-Theoretic Topology, K. Kunen and J. Vaughan, eds., Chapter 4, North-Holland, Amsterdam, 1984, pp.169-200. Zbl0769.54008MR0776623

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.