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Insufficiency of the hyperspace

Published online by Cambridge University Press:  24 October 2008

John R. Isbell
John R. Isbell
Affiliation:
Case Institute of Technology
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I asserted in (1), p. 35, that the topology of the hyperspace H(μX)of a uniform space μX determines the uniformity μ, and sketched a proof. D. Hammond Smith has shown (2) that the proof indicated is wrong, even for metric spaces. However, Smith proved the assertion for metric spaces and some others, and left open the question whether it was true in general. This note gives a counter example. I should note that I began with a transfinite construction based on Smith's example; whether that would have worked or not, the following construction is simpler.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 62 , Issue 4 , October 1966 , pp. 685 - 686
Copyright
Copyright © Cambridge Philosophical Society 1966

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References

REFERENCES

(1)Isbell, J. R.Uniform spaces. Providence, Rhode Island. American Math. Soc. (1964).Google Scholar
(2)Smith, D. H.Hyperspaces of a uniformizable space. Proc. Cambridge Phil. Soc. 62 (1966), 25–28.CrossRefGoogle Scholar