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Decomposition of Metric Spaces with a 0-Dimensional Set of Non-Degenerate Elements

Published online by Cambridge University Press:  20 November 2018

Jack W. Lamoreaux
Jack W. Lamoreaux*
Affiliation:
University of Alberta, Edmonton, Alberta Brigham Young University, Provo, Utah
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Various conditions under which an upper semi-continuous (u.s.-c.) decomposition of E3 yields E3 as its decomposition space have been given by Armentrout (1; 2; 5), Bing (7; 8), Lambert (13), McAuley (14), Smythe (17), and Wardwell (18). If the projection of the non-degenerate elements is 0-dimensional in the decomposition space, then “shrinking” or “Condition B” (6) has proven particularly useful.

In this paper we shall investigate monotone u.s.-c. decompositions of a locally compact connected metric space M, where the projection of the nondegenerate elements is 0-dimensional. We show in Theorem 1 that each open covering of the non-degenerate elements of a 0-dimensional decomposition has a locally finite refinement.

In § 5, we use Theorem 1 to investigate the following question which is similar to one raised by Bing (11, p. 19): Let G, G′, and G″ be decompositions of M such that the non-degenerate elements of G are those of G′ together with those of G′.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 202 - 216
Copyright
Copyright © Canadian Mathematical Society 1969

References

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