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ENTCS
2008

Connectivity Properties of Dimension Level Sets

9 years 12 months ago
Connectivity Properties of Dimension Level Sets
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 DIMI 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 totally disconnected. In contrast, we show that the sets DIM[0,1] and DIM[n-1,n] are path-connected. Our proof of this fact uses geometric properties of Kolmogorov complexity in Euclidean space.
Jack H. Lutz, Klaus Weihrauch
Added 10 Dec 2010
Updated 10 Dec 2010
Type Journal
Year 2008
Where ENTCS
Authors Jack H. Lutz, Klaus Weihrauch
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