A relatively free topological group that is not varietal free

Vladimir Pestov; Dmitri Shakhmatov

Colloquium Mathematicae (1998)

  • Volume: 77, Issue: 1, page 1-8
  • ISSN: 0010-1354

Abstract

top
Answering a 1982 question of Sidney A. Morris, we construct a topological group G and a subspace X such that (i) G is algebraically free over X, (ii) G is relatively free over X, that is, every continuous mapping from X to G extends to a unique continuous endomorphism of G, and (iii) G is not a varietal free topological group on X in any variety of topological groups.

How to cite

top

Pestov, Vladimir, and Shakhmatov, Dmitri. "A relatively free topological group that is not varietal free." Colloquium Mathematicae 77.1 (1998): 1-8. <http://eudml.org/doc/210574>.

@article{Pestov1998,
abstract = {Answering a 1982 question of Sidney A. Morris, we construct a topological group G and a subspace X such that (i) G is algebraically free over X, (ii) G is relatively free over X, that is, every continuous mapping from X to G extends to a unique continuous endomorphism of G, and (iii) G is not a varietal free topological group on X in any variety of topological groups.},
author = {Pestov, Vladimir, Shakhmatov, Dmitri},
journal = {Colloquium Mathematicae},
keywords = {relatively free topological group; variety of topological groups; free zero-dimensional topological group; varietal free topological group; topological groups; variety; algebraically free},
language = {eng},
number = {1},
pages = {1-8},
title = {A relatively free topological group that is not varietal free},
url = {http://eudml.org/doc/210574},
volume = {77},
year = {1998},
}

TY - JOUR
AU - Pestov, Vladimir
AU - Shakhmatov, Dmitri
TI - A relatively free topological group that is not varietal free
JO - Colloquium Mathematicae
PY - 1998
VL - 77
IS - 1
SP - 1
EP - 8
AB - Answering a 1982 question of Sidney A. Morris, we construct a topological group G and a subspace X such that (i) G is algebraically free over X, (ii) G is relatively free over X, that is, every continuous mapping from X to G extends to a unique continuous endomorphism of G, and (iii) G is not a varietal free topological group on X in any variety of topological groups.
LA - eng
KW - relatively free topological group; variety of topological groups; free zero-dimensional topological group; varietal free topological group; topological groups; variety; algebraically free
UR - http://eudml.org/doc/210574
ER -

References

top
  1. [1] A. V. Arhangel'skiĭ [A. V. Arkhangel'skiĭ], Any topological group is a quotient group of a zero-dimensional topological group, Soviet. Math. Dokl. 23 (1981), 615-618. 
  2. [2] A. V. Arhangel'skiĭ [A. V. Arkhangel'skiĭ], Classes of topological groups, Russian Math. Surveys 36 (1981), 151-174. 
  3. [3] M. S. Brooks, S. A. Morris and S. A. Saxon, Generating varieties of topological groups, Proc. Edinburgh Math. Soc. 18 (1973), 191-197. Zbl0263.22002
  4. [4] W. W. Comfort and J. van Mill, On the existence of free topological groups, Topology Appl. 29 (1988), 245-265. 
  5. [5] R. Engelking, General Topology, PWN, Warszawa, 1977. 
  6. [6] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, 2nd ed. Springer, 1979. Zbl0416.43001
  7. [7] K. H. Hofmann, An essay on free compact groups, in: Lecture Notes in Math. 915, Springer, 1982, 171-197. 
  8. [8] H. J. K. Junnila, Stratifiable pre-images of topological spaces, in: Topology (Budapest 1978), Colloq. Math. Soc. János Bolyai 23, North-Holland, 1980, 689-703. 
  9. [9] S. Mac Lane, Categories for the Working Mathematician, Grad. Texts in Math. 5, Springer, 1971. 
  10. [10] A. A. Markov, On free topological groups, Dokl. Akad. Nauk SSSR 31 (1941), 299-301 (in Russian). 
  11. [11] A. A. Markov, Three papers on topological groups, Amer. Math. Soc. Transl. 30 (1950), 120 pp. 
  12. [12] S. A. Morris, Varieties of topological groups, Bull. Austral. Math. Soc. 1 (1969), 145-160. Zbl0172.31404
  13. [13] S. A. Morris, Varieties of topological groups and left adjoint functor, J. Austral. Math. Soc. 16 (1973), 220-227. Zbl0274.22003
  14. [14] S. A. Morris, Varieties of topological groups. A survey, Colloq. Math. 46 (1982), 147-165. Zbl0501.22002
  15. [15] S. A. Morris, Free abelian topological groups, in: Categorical Topology (Toledo, Ohio, 1983), Heldermann, 1984, 375-391. 
  16. [16] H. Neumann, Varieties of Groups, Ergeb. Math. Grenzgeb. 37, Springer, Berlin, 1967. Zbl0149.26704
  17. [17] V. G. Pestov, Neighbourhoods of unity in free topological groups, Moscow Univ. Math. Bull. 40 (1985), 8-12. Zbl0592.22002
  18. [18] V. G. Pestov, Universal arrows to forgetful functors from categories of topological algebra, Bull. Austral. Math. Soc. 48 (1993), 209-249. Zbl0784.18002
  19. [19] P. Samuel, On universal mappings and free topological groups, Bull. Amer. Math. Soc. 54 (1948), 591-598. Zbl0031.41702
  20. [20] D. B. Shakhmatov, Zerodimensionality of free topological groups and topological groups with noncoinciding dimensions, Bull. Polish Acad. Sci. Math. 37 (1989), 497-506. Zbl0759.54023
  21. [21] D. B. Shakhmatov, Imbeddings into topological groups preserving dimensions, Topology Appl. 36 (1990), 181-204. Zbl0709.22001

NotesEmbed ?

top

You must be logged in to post comments.