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A counter-example in uniformity theory

Published online by Cambridge University Press:  24 October 2008

A. J. Ward
A. J. Ward
Affiliation:
Emmanuel College, Cambridge
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Extract

In a recent paper, Smith ((l)), considering a statement or conjecture of Isbell ((2)), proved that the Hausdorff uniformities arising from two different uniformities on the same set must induce different topologies on the ‘hyperspace’ of subsets, except possibly in one case which remained open, namely, when the two given uniformities induce the same proximity and neither is precompact. I now give an example to show that different uniformities can (in this case) induce the same topology on the hyperspace (in the example one of the uniformities is metric). This disproves Isbell's conjecture.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 62 , Issue 2 , April 1966 , pp. 207 - 208
Copyright
Copyright © Cambridge Philosophical Society 1966

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References

REFERENCES

(1)Smith, D. H.Proc. Cambridge Philos. Soc. 62 (1966),CrossRefGoogle Scholar
(2)Isbell, J. R.Uniform spaces (Math. Surveys 12), Amer. Math. Soc. 1964. p. 35. Ex. 17.Google Scholar