Extensions of topological and semitopological groups and the product operation

Aleksander V. Arhangel'skii; Miroslav Hušek

Commentationes Mathematicae Universitatis Carolinae (2001)

  • Volume: 42, Issue: 1, page 173-186
  • ISSN: 0010-2628

Abstract

top
The main results concern commutativity of Hewitt-Nachbin realcompactification or Dieudonné completion with products of topological groups. It is shown that for every topological group G that is not Dieudonné complete one can find a Dieudonné complete group H such that the Dieudonné completion of G × H is not a topological group containing G × H as a subgroup. Using Korovin’s construction of G δ -dense orbits, we present some examples showing that some results on topological groups are not valid for semitopological groups.

How to cite

top

Arhangel'skii, Aleksander V., and Hušek, Miroslav. "Extensions of topological and semitopological groups and the product operation." Commentationes Mathematicae Universitatis Carolinae 42.1 (2001): 173-186. <http://eudml.org/doc/248816>.

@article{Arhangelskii2001,
abstract = {The main results concern commutativity of Hewitt-Nachbin realcompactification or Dieudonné completion with products of topological groups. It is shown that for every topological group $G$ that is not Dieudonné complete one can find a Dieudonné complete group $H$ such that the Dieudonné completion of $G\times H$ is not a topological group containing $G\times H$ as a subgroup. Using Korovin’s construction of $G_\delta $-dense orbits, we present some examples showing that some results on topological groups are not valid for semitopological groups.},
author = {Arhangel'skii, Aleksander V., Hušek, Miroslav},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {topological group; Dieudonné completion; PT-group; realcompactness; Moscow space; $C$-embedding; product; topological group; Dieudonné completion; realcompactness; product},
language = {eng},
number = {1},
pages = {173-186},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Extensions of topological and semitopological groups and the product operation},
url = {http://eudml.org/doc/248816},
volume = {42},
year = {2001},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
AU - Hušek, Miroslav
TI - Extensions of topological and semitopological groups and the product operation
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2001
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 42
IS - 1
SP - 173
EP - 186
AB - The main results concern commutativity of Hewitt-Nachbin realcompactification or Dieudonné completion with products of topological groups. It is shown that for every topological group $G$ that is not Dieudonné complete one can find a Dieudonné complete group $H$ such that the Dieudonné completion of $G\times H$ is not a topological group containing $G\times H$ as a subgroup. Using Korovin’s construction of $G_\delta $-dense orbits, we present some examples showing that some results on topological groups are not valid for semitopological groups.
LA - eng
KW - topological group; Dieudonné completion; PT-group; realcompactness; Moscow space; $C$-embedding; product; topological group; Dieudonné completion; realcompactness; product
UR - http://eudml.org/doc/248816
ER -

References

top
  1. Arhangel'skii A.V., Functional tightness, Q -spaces, and τ -embeddings, Comment. Math. Univ. Carolinae 24:1 (1983), 105-120. (1983) MR0703930
  2. Arhangel'skii A.V., Moscow spaces, Pestov-Tkačenko Problem, and C -embeddings, Comment. Math. Univ. Carolinae 41:3 (2000), 585-595. (2000) Zbl1038.54013MR1795087
  3. Arhangel'skii A.V., On power homogeneous spaces, Proc. Yokohama Topology Conference, Japan, 1999 (to appear in Topology Appl. (2001)). Zbl0994.54028MR1919289
  4. Comfort W.W., Negrepontis S., Continuous functions on products with strong topologies, Proc. 3rd Prague Top. Symp. 1971 (Academia, Prague 1972), pp.89-92. Zbl0308.54005MR0358656
  5. Comfort W.W., Ross K.A., Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16 (1966), 483-496. (1966) Zbl0214.28502MR0207886
  6. Ellis R., Locally compact transformation groups, Duke Math. J. 24 (1957), 119-125. (1957) Zbl0079.16602MR0088674
  7. Engelking R., General Topology, PWN, Warszawa, 1977. Zbl0684.54001MR0500780
  8. Hušek M., Realcompactness of function spaces and υ ( P × Q ) , General Topology and Appl. 2 (1972), 165-179. (1972) MR0307181
  9. Hušek M., Pelant J., Extensions and restrictions in products of metric spaces, Topology Appl. 25 (1987), 245-252. (1987) MR0889869
  10. Hušek M., de Vries J., Preservation of products by functors close to reflectors, Topology Appl. 27 (1987), 171-189. (1987) MR0911690
  11. Korovin A.V., Continuous actions of pseudocompact groups and axioms of topological group, Comment. Math. Univ. Carolinae 33 (1992), 335-343. (1992) Zbl0786.22002MR1189665
  12. Pašenkov V.V., Extensions of compact spaces, Soviet Math. Dokl. 15:1 (1974), 43-47. (1974) 
  13. Pestov V.G., Tkačenko M.G., Problem 3.28, in Unsolved Problems of Topological Algebra, Academy of Science, Moldova, Kishinev, ``Shtiinca'' 1985, p.18. 
  14. Reznichenko E.A., Extensions of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, Topology Appl. 59 (1994), 233-244. (1994) MR1299719
  15. Roelke W., Dierolf S., Uniform Structures on Topological Groups and Their Quotients, McGraw-Hill, New York, 1981. 
  16. Shirota T., A class of topological spaces, Osaka Math. J. 4 (1952), 23-40. (1952) Zbl0047.41704MR0050872
  17. Tkačenko M.G., Subgroups, quotient groups, and products of R -factorizable groups, Topology Proc. 16 (1991), 201-231. (1991) MR1206464
  18. Uspenskij V.V., Topological groups and Dugundji spaces, Matem. Sb. 180:8 (1989), 1092-1118. (1989) MR1019483

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.