Moscow spaces, Pestov-Tkačenko Problem, and C -embeddings

Aleksander V. Arhangel'skii

Commentationes Mathematicae Universitatis Carolinae (2000)

  • Volume: 41, Issue: 3, page 585-595
  • ISSN: 0010-2628

Abstract

top
We show that there exists an Abelian topological group G such that the operations in G cannot be extended to the Dieudonné completion μ G of the space G in such a way that G becomes a topological subgroup of the topological group μ G . This provides a complete answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions to the equation υ X × υ Y = υ ( X × Y ) . The key role in the approach belongs to the notion of Moscow space which turns out to be very useful in the theory of C -embeddings and interacts especially well with homogeneity.

How to cite

top

Arhangel'skii, Aleksander V.. "Moscow spaces, Pestov-Tkačenko Problem, and $C$-embeddings." Commentationes Mathematicae Universitatis Carolinae 41.3 (2000): 585-595. <http://eudml.org/doc/248634>.

@article{Arhangelskii2000,
abstract = {We show that there exists an Abelian topological group $G$ such that the operations in $G$ cannot be extended to the Dieudonné completion $\mu G$ of the space $G$ in such a way that $G$ becomes a topological subgroup of the topological group $\mu G$. This provides a complete answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions to the equation $\upsilon X\times \upsilon Y=\upsilon (X\times Y)$. The key role in the approach belongs to the notion of Moscow space which turns out to be very useful in the theory of $C$-embeddings and interacts especially well with homogeneity.},
author = {Arhangel'skii, Aleksander V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Moscow space; Dieudonné completion; Hewitt-Nachbin completion; $C$-embedding; $G_\delta $-dense set; topological group; Souslin number; tightness; canonical open set; Dieudonné completion; Rajkov completion; Hewitt-Nachbin completion; -embedding; topological group},
language = {eng},
number = {3},
pages = {585-595},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Moscow spaces, Pestov-Tkačenko Problem, and $C$-embeddings},
url = {http://eudml.org/doc/248634},
volume = {41},
year = {2000},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
TI - Moscow spaces, Pestov-Tkačenko Problem, and $C$-embeddings
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2000
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 41
IS - 3
SP - 585
EP - 595
AB - We show that there exists an Abelian topological group $G$ such that the operations in $G$ cannot be extended to the Dieudonné completion $\mu G$ of the space $G$ in such a way that $G$ becomes a topological subgroup of the topological group $\mu G$. This provides a complete answer to a question of V.G. Pestov and M.G. Tkačenko, dating back to 1985. We also identify new large classes of topological groups for which such an extension is possible. The technique developed also allows to find many new solutions to the equation $\upsilon X\times \upsilon Y=\upsilon (X\times Y)$. The key role in the approach belongs to the notion of Moscow space which turns out to be very useful in the theory of $C$-embeddings and interacts especially well with homogeneity.
LA - eng
KW - Moscow space; Dieudonné completion; Hewitt-Nachbin completion; $C$-embedding; $G_\delta $-dense set; topological group; Souslin number; tightness; canonical open set; Dieudonné completion; Rajkov completion; Hewitt-Nachbin completion; -embedding; topological group
UR - http://eudml.org/doc/248634
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., On a Theorem of W.W. Comfort and K.A. Ross, Comment. Math. Univ. Carolinae 40:1 (1999), 133-151. (1999) MR1715207
  3. Arhangel'skii A.V., Topological groups and C -embeddings, submitted, 1999. Zbl0984.54018
  4. Blair R.L., Spaces in which special sets are z -embedded, Canad. J. Math. 28:4 (1976), 673-690. (1976) Zbl0359.54009MR0420542
  5. Blair R.L., Hager A.W., Notes on the Hewitt realcompactification of a product, Gen. Topol. and Appl. 5 (1975), 1-8. (1975) Zbl0323.54021MR0365496
  6. Comfort W.W., On the Hewitt realcompactification of the product space, Trans. Amer. Math. Soc. 131 (1968), 107-118. (1968) MR0222846
  7. Comfort W.W., Negrepontis S., Extending continuous functions on X × Y to subsets of β X × β Y , Fund. Math. 59 (1966), 1-12. (1966) Zbl0185.26304MR0200896
  8. Comfort W.W., Ross K.A., Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16:3 (1966), 483-496. (1966) Zbl0214.28502MR0207886
  9. Engelking R., General Topology, PWN, Warszawa, 1977. Zbl0684.54001MR0500780
  10. Frolík Z., The topological product of two pseudocompact spaces, Czechoslovak Math. J. 10 (1960), 339-349. (1960) MR0116304
  11. Gillman L., Jerison M., Rings of Continuous Functions, Princeton, 1960. Zbl0327.46040MR0116199
  12. Glicksberg I., Stone-Čech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369-382. (1959) Zbl0089.38702MR0105667
  13. Hewitt E., Rings of real-valued continuous functions 1., Trans. Amer. Math. Soc. 64 ((1948)), 45-99. ((1948)) MR0026239
  14. Hušek M., Realcompactness of function spaces and υ ( P × Q ) , Gen. Topol. and Appl. 2 (1972), 165-179. (1972) MR0307181
  15. Pestov V.G., Tkačenko M.G., Problem 3 . 28 , in: Unsolved Problems of Topological Algebra, Academy of Science, Moldova, Kishinev, "Shtiinca" 1985, p.18. 
  16. Roelke W., Dierolf S., Uniform Structures on Topological Groups and Their Quotients, McGraw-Hill, New York, 1981. 
  17. Stchepin E.V., On κ -metrizable spaces, Izv. Akad. Nauk SSSR, Ser. Matem. 43:2 (1979), 442-478. (1979) MR0534603
  18. Terada T., Note on z -, C * -, and C -embedded subspaces of products, Sci. Rep. Tokyo Kyoiku Daigaku Sect. A 13 (1975), 129-132. (1975) Zbl0333.54008MR0391005
  19. Tkačenko M.G., The notion of o -tightness and C -embedded subspaces of products, Topology Appl. 15 (1983), 93-98. (1983) MR0676970
  20. Tkačenko M.G., Subgroups, quotient groups, and products of R -factorizable groups, Topology Proc. 16 (1991), 201-231. (1991) MR1206464
  21. Tkačenko M.G., Introduction to Topological Groups, Topology Appl. 86:3 (1998), 179-231. (1998) MR1623960
  22. 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.