Geodesic
Decomposition of Metric Spaces with a 0-Dimensional Set of Non-Degenerate Elements
Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 202-216

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Various conditions under which an upper semi-continuous (u.s.-c.) decomposition of E 3 yields E 3 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′. 
Lamoreaux, Jack W. Decomposition of Metric Spaces with a 0-Dimensional Set of Non-Degenerate Elements. Canadian journal of mathematics, Tome 21 (1969) no. 1, pp. 202-216. doi: 10.4153/CJM-1969-020-7
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