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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
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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

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