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Uniform Spaces as Nice Images of Nice Uniform and Metric Spaces(1)

Published online by Cambridge University Press:  20 November 2018

Richard Willmott
Richard Willmott*
Affiliation:
Department of Mathematics Queen's University, Kingston, Ontario K7L 3N6
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The classical theorem that a complete separable metric space is the image under a one-to-one continuous function of a closed subset of the irrational numbers has been extended in two directions, the first leading to various characterizations in descriptive set theory of Borel and analytic sets or generalizations of them as continuous images of certain subsets of the irrationals, or generalizations of them (see, e.g. [3] and references cited there; [4]; [6]). The second direction originates in the observation that a closed subset of the irrationals is a complete 0-dimensional metric space (under a suitable metric), and leads to the general question asked by Alexandroff [1], "Which spaces can be represented as images of 'nice' (e.g. metric, 0-dimensional) spaces under 'nice' [e.g. one-to-one, open, closed, perfect] continuous mappings?" (See, e.g. [7], [9] and the survey articles [1], [2] and [11].)

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 23 , Issue 1 , 01 March 1980 , pp. 1 - 9
Copyright
Copyright © Canadian Mathematical Society 1980

Footnotes

(1)

This work was supported by the National Research Council of Canada.

References

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