Some Upper Semi-Continuous Decompositions of E 3 into E 3

Abstract

One result of this paper is to generalize a theorem due to R. H. Bing 11, Theorem 1]. Although Bing's techniques supplied some ideas here, the author looks at certain coverings of the non-degenerate elements of an upper semi-continuous decomposition of E3, while Bing concentrates essentially on the elements. These observations concerning special coverings of the non-degenerate point-like elements lead to other decomposition theorems. One of them solves a special case of a problem proposed by R. H. Bing and Deane Montgomery. Perhaps this theorem may be used to solve the problem completely. It is stated as follows. Suppose that G is an upper semi-continuous collection of straight line intervals and points filling up E3. Is the decomposition space homeomorphic to E3? If each element of the collection H of all non-degenerate elements of G is parallel to some one of a countably infinite number of lines, then Theorem 3 below states that the decomposition space is topologically E3. Such a result is not unexpected and generalizations of it are also obtained. However, an example of Bing [2] modified by Fort [6] gives a decomposition of E3 into points and polygonal arcs, each having exactly two bends occurring in two fixed planes, which fails to be topologically E3. Thus, one finds that simple decompositions of E3 may not yield an obvious hyperspace. The author is deeply indebted to R. H. Bing for many valuable suggestions and ideas stimulated by his papers. Theorem 1 below generalizes a theorem of Bing. The collection of nondegenerate elements may fail to be countable as required in [1]. It is interesting to note that in the example of Bing (modified by Fort), men-