On a theorem of W.W. Comfort and K.A. Ross

Aleksander V. Arhangel'skii

Commentationes Mathematicae Universitatis Carolinae (1999)

  • Volume: 40, Issue: 1, page 133-151
  • ISSN: 0010-2628

Abstract

top
A well known theorem of W.W. Comfort and K.A. Ross, stating that every pseudocompact group is C -embedded in its Weil completion [5] (which is a compact group), is extended to some new classes of topological groups, and the proofs are very transparent, short and elementary (the key role in the proofs belongs to Lemmas 1.1 and 4.1). In particular, we introduce a new notion of canonical uniform tightness of a topological group G and prove that every G δ -dense subspace Y of a topological group G , such that the canonical uniform tightness of G is countable, is C -embedded in G .

How to cite

top

Arhangel'skii, Aleksander V.. "On a theorem of W.W. Comfort and K.A. Ross." Commentationes Mathematicae Universitatis Carolinae 40.1 (1999): 133-151. <http://eudml.org/doc/248387>.

@article{Arhangelskii1999,
abstract = {A well known theorem of W.W. Comfort and K.A. Ross, stating that every pseudocompact group is $C$-embedded in its Weil completion [5] (which is a compact group), is extended to some new classes of topological groups, and the proofs are very transparent, short and elementary (the key role in the proofs belongs to Lemmas 1.1 and 4.1). In particular, we introduce a new notion of canonical uniform tightness of a topological group $G$ and prove that every $G_\delta $-dense subspace $Y$ of a topological group $G$, such that the canonical uniform tightness of $G$ is countable, is $C$-embedded in $G$.},
author = {Arhangel'skii, Aleksander V.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {topological group; pseudocompact; Frechet-Urysohn; $G_\delta $-dense; $C$-embedded; Moscow space; canonical uniform tightness; Hewitt completion; Rajkov completion; bounded set; extremally disconnected; normal space; $k_1$-space; topological group; pseudocompact space; C-embedding; extremally disconnected space},
language = {eng},
number = {1},
pages = {133-151},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {On a theorem of W.W. Comfort and K.A. Ross},
url = {http://eudml.org/doc/248387},
volume = {40},
year = {1999},
}

TY - JOUR
AU - Arhangel'skii, Aleksander V.
TI - On a theorem of W.W. Comfort and K.A. Ross
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1999
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 40
IS - 1
SP - 133
EP - 151
AB - A well known theorem of W.W. Comfort and K.A. Ross, stating that every pseudocompact group is $C$-embedded in its Weil completion [5] (which is a compact group), is extended to some new classes of topological groups, and the proofs are very transparent, short and elementary (the key role in the proofs belongs to Lemmas 1.1 and 4.1). In particular, we introduce a new notion of canonical uniform tightness of a topological group $G$ and prove that every $G_\delta $-dense subspace $Y$ of a topological group $G$, such that the canonical uniform tightness of $G$ is countable, is $C$-embedded in $G$.
LA - eng
KW - topological group; pseudocompact; Frechet-Urysohn; $G_\delta $-dense; $C$-embedded; Moscow space; canonical uniform tightness; Hewitt completion; Rajkov completion; bounded set; extremally disconnected; normal space; $k_1$-space; topological group; pseudocompact space; C-embedding; extremally disconnected space
UR - http://eudml.org/doc/248387
ER -

References

top
  1. Arhangel'skii A.V., Classes of topological groups, Russian Math. Surveys 36:3 (1981), 151-174. (1981) MR0622722
  2. Arhangel'skii A.V., Functional tightness, Q -spaces, and τ -embeddings, Comment. Math. Univ. Carolinae 24:1 (1983), 105-120. (1983) MR0703930
  3. Arhangel'skii A.V., Ponomarev V.I., Fundamentals of General Topology in Problems and Exercises, D. Reidel Publ. Co., Dordrecht-Boston, Mass., 1984. MR0785749
  4. Chigogidze A.C., On κ -metrizable spaces, Russian Math. Surveys 37 (1982), 209-210. (1982) Zbl0503.54012MR0650791
  5. Comfort W.W., Ross K.A., Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16:3 (1966), 483-496. (1966) Zbl0214.28502MR0207886
  6. Comfort W.W., Trigos-Arrieta F.J., Locally pseudocompact topological groups, Topology Appl. 62:3 (1995), 263-280. (1995) Zbl0828.22003MR1326826
  7. Engelking R., General Topology, PWN, Warszawa, 1977. Zbl0684.54001MR0500780
  8. Glicksberg I., Stone-Čech compactifications of products, Trans. Amer. Math. Soc. 90 (1959), 369-382. (1959) Zbl0089.38702MR0105667
  9. Hernandez S., Sanchis M., G δ -open functionally bounded subsets in topological groups, Topology Appl. 53:3 (1993), 288-299. (1993) MR1254873
  10. Hernandez S., Sanchis M., Tkachenko M., Bounded sets in spaces and topological groups, preprint, 1997. Zbl0979.54037
  11. Hušek M., Productivity of properties of topological groups, Topology Appl. 44 (1992), 189-196. (1992) MR1173257
  12. Michael E., A quintuple quotient quest, Gen. Topol. and Appl. 2:1 (1972), 91-138. (1972) Zbl0238.54009MR0309045
  13. Pontryagin L.S., Continuous Groups, Princeton Univ. Press, Princeton, N.J., 1956. Zbl0659.22001
  14. Roelke W., Dierolf S., Uniform Structures on Topological Groups and their Quotients, McGraw-Hill, New York, 1981. 
  15. Tkachenko M.G., Subgroups, quotient groups, and products of R -factorizable groups, Topology Proc. 16 (1991), 201-231. (1991) MR1206464
  16. Tkachenko M.G., Compactness type properties in topological groups, Czech. Math. J. 38 (1988), 324-341. (1988) Zbl0664.54006MR0946302
  17. Tkachenko M.G., The notion of o -tightness and C -embedded subspaces of products, Topology Appl. 15 (1983), 93-98. (1983) Zbl0509.54013MR0676970
  18. Uspenskij V.V., Topological groups and Dugundji spaces, Matem. Sb. 180:8 (1989), 1092-1118. (1989) MR1019483
  19. de Vries J., Pseudocompactness and the Stone-Čech compactification for topological groups, Nieuw Archef voor Wiskunde 23:3 (1975), 35-48. (1975) Zbl0296.22003MR0401978

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.