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A generalization of isometries to uniform spaces

Published online by Cambridge University Press:  24 October 2008

F. Rhodes
F. Rhodes
Affiliation:
The UniversityLiverpool
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Extract

Isomorphisms are, in many ways, the generalizations of isometrics to uniform spaces. Yet some theorems on isometries of metric spaces only generalize to uniform spaces in terms of more restricted transformations of the uniform space. In § 1, in the course of a discussion of a theorem on transitive groups of automorphisms, we define such a transformation and call it an isobasism. It appears that in many respects isobasisms, rather than isomorphisms, are the generalizations of isometries to uniform spaces. The results of Freudenthal and Hurewicz (7) on contractions, expansions and isometries of totally bounded metric spaces are generalized, in § 2, to contractions, expansions and isobasisms of totally bounded uniform spaces. These results, together with generalizations of some theorems of Eilenberg (6) on compact groups of homeomorphisms of metric spaces which are obtained in §3, give a characterization of isobasisms. The language of Bourbaki (2,3,4) is used throughout this note.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 52 , Issue 3 , July 1956 , pp. 399 - 405
Copyright
Copyright © Cambridge Philosophical Society 1956

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References

REFERENCES

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