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Invariant metrics on free topological groups

Published online by Cambridge University Press:  17 April 2009

Sidney A. Morris  and
H.B. Thompson
Sidney A. Morris
Affiliation:
School of Mathematics, University of New South Wales, Kensington, New South Wales;
H.B. Thompson
Affiliation:
School of Mathematical Sciences, Flinders University, Bedford Park, South Australia.
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Abstract

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For a completely regular space X, G(X) denotes the free topological group on X in the sense of Graev. Graev proves the existence of G(X) by showing that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). It is natural to ask if the topology given by these two-sided invariant pseudo-metrics on G(X) is precisely the free topological group topology on G(X). If X has the discrete topology the answer is clearly in the affirmative. It is shown here that if X is not totally disconnected then the answer is always in the negative.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 9 , Issue 1 , August 1973 , pp. 83 - 88
Copyright
Copyright © Australian Mathematical Society 1973

References

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