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This paper initiates the study of sets in Euclidean space Rn(n⩾2) that are defined in terms of the dimensions of their elements. Specifically, given an interval I⊆[0,1], we are interested in the connectivity properties of the set DIM I consisting of all points in Rn whose (constructive Hausdorff) dimensions lie in the interval I. It is easy to see that the sets DIM[0,1) and DIM(n−1,n] are...
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