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A theorem about hyperspaces

Published online by Cambridge University Press:  24 October 2008

J. L. Hursch Jr
J. L. Hursch Jr
Affiliation:
University of Vermont
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Extract

Recently Isbell(4) and Ward (6) have given examples of pairs of uniformities inducing the same topology on the hyperspace of subsets of a uniform space. Smith (5) has shown that, if two uniformities induce the same topology on the hyperspace, they must be equal in proximity. The author, in (2), has introduced a relation, ‘height’, between uniformities which is in a sense dual to proximity. It is the purpose of this paper to show that, if two uniformities are equal in proximity and height, then they induce the same topology on the hyperspace. Although Ward's uniformities are not equal in height, an example of two distinct uniformities equal in proximity and height is given in (3). In fact, a slight modification of that example will give an example of a countably infinite number of distinct uniformities, all of which are equal in proximity and height.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 63 , Issue 3 , July 1967 , pp. 597 - 599
Copyright
Copyright © Cambridge Philosophical Society 1967

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References

REFERENCES

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