Hostname: page-component-7479d7b7d-k7p5g Total loading time: 0 Render date: 2024-07-09T01:24:25.886Z Has data issue: false hasContentIssue false
  • English
  • Français

Local invariance of free topological groups

Published online by Cambridge University Press:  20 January 2009

M. S. Khan ,
Sidney A. Morris  and
Peter Nickolas
M. S. Khan
Affiliation:
Department of Mathematics, King Abdul Aziz University, P.O. Box 9028, Jeddah-21413, Saudi Arabia
Sidney A. Morris
Affiliation:
Department of Mathematics, La Trobe University, Bundoora 3083, Australia
Peter Nickolas
Affiliation:
Department of Computing Science, University of Wollongong, Wollongong, 2500, Australia
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1948, M. I. Graev [2] proved that the free topological group on a completely regular Hausdorff space is Hausdorff, by showing that the free group admits a certain locally invariant Hausdorff group topology. It is natural to ask if Graev's locally invariant topology is the free topological group topology. If X has the discrete topology, the answer is clearly in the affirmative. In 1973, Morris-Thompson [6] showed that if X is not totally disconnected then the answer is negative. Nickolas [7] showed that this is also the case if X has any (non-trivial) convergent sequence (for example, if X is any non-discrete metric space). Recently, Fay and Smith Thomas handled the case when X has a completely regular Hausdorff quotient space which has an infinite compact subspace (or more particularly a non-trivial convergent sequence).(Fay-Smith Thomas observe that their class of spaces includes some but not all those dealt with by Morris-Thompson.)

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 29 , Issue 1 , February 1986 , pp. 1 - 5
Copyright
Copyright © Edinburgh Mathematical Society 1986

References

REFERENCES

1.Fay, Temple H. and Smith Thomas, Barbara V., Free topological groups are almost never locally Invariant, Math. Colloq. Univ. Capetown 13 (1984), 1–8.Google Scholar
2.Graev, M. I., Free topological groups, Izvestiya Akad. Nauk SSSR. Ser. Mat. 12 (1948), 279–324. English trans.:Google Scholar
Amer. Math. Soc. Transl. 35 (1951); reprintGoogle Scholar
Amer. Math. Soc. Transl. (1) 8 (1962), 305–364.Google Scholar
3.Mack, John, Morris, Sidney A. and Ordman, Edward T., Free topological groups and the projective dimension of a locally compact abelian group, Proc. Amer. Math. Soc. 40 (1973), 303–308.Google Scholar
4.Miller, Arnold W.On box products, Topology Applic. 14 (1982), 313–317.CrossRefGoogle Scholar
5.Morris, Sidney A. and Nickolas, Peter, Locally Invariant topologies on free groups, Pacific J. Math. 103 (1982), 523–537.Google Scholar
6.Morris, Sidney A. and Thompson, H. B., Invariant metrics on free topological groups, Bull. Austral. Math. Soc. 9 (1973), 83–88.CrossRefGoogle Scholar
7.Nickolas, Peter, Free topological groups and free products of topological groups (Ph.D. thesis, University of New South Wales, Australia, 1976).Google Scholar
8.Nummela, E., Uniform free topological groups and Samuel compactifications, Topology Applic. 13 (1982), 77–83.CrossRefGoogle Scholar