A generalization of the dog bone space to 𝐸ⁿ

In this paper we construct an upper semicontinuous decomposition of En (n_3) into points and tame arcs such that the associated decomposition space is distinct from En. The purpose of this paper is to construct an upper semicontinuous decomposition G of En, n>3, into points and tame arcs such that the decomposition space EnIG is topologically distinct from En. For n=3, the example is a modification of R. H. Bing's dog bone space [2]. Unlike the dog bone space, it is not difficult to distinguish our decomposition spaces from Euclidean space. The main idea for the construction of these decomposition spaces was communicated to us by R. D. Anderson during his visit to the University of Texas in March, 1972. Anderson's idea was roughly the following: Take two disjoint wild Cantor sets in En. To each point in one Cantor set correspond a unique point in the other Cantor set and join the two points with an arc that is locally polygonal modulo its end points. The collection of arcs thus obtained can be constructed so that its union is homeomorphic to the product of a Cantor set and an arc. Such arcs for n>4 are tame and, hopefully, if the Cantor sets are wild enough and the pairings are chosen cleverly, the resulting upper semicontinuous decomposition of En will have a decomposition space distinct from En. The difficulty of proving that such decomposition spaces are not En lies in finding a topological property of En not shared by the decomposition spaces. We use the following elementary property of En. THEOREM 1. If C is a Cantor set in En (n _ 3), U is an open set containing C, andf and g are maps from a 2-cell D into En, then there exist maps f' and g' from D into En such that f'lf-1(EnU)=f If`1(EnU), g'lg-1(En U) =gjg`(EnU), fI(f(U)) c U, g'(g-(U)) c U and f'(D ng'(D) nx C_ = Received by the editors July 24, 1972. AMS (MOS) subject class/ifcations (1970). Primary 57A15, 57A10, 54A30; Secondary 57A35, 57A45.