Connectivity Properties of Dimension Level Sets | Zendy

Jack H. Lutz | Zendy; Klaus Weihrauch | Zendy

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Connectivity Properties of Dimension Level Sets

Author(s) -

Jack H. Lutz,

Klaus Weihrauch

Publication year - 2008

Publication title -

electronic notes in theoretical computer science

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 0.242

H-Index - 60

ISSN - 1571-0661

DOI - 10.1016/j.entcs.2008.03.022

Subject(s) - mathematics , euclidean space , constructive proof , dimension (graph theory) , euclidean geometry , interval (graph theory) , combinatorics , space (punctuation) , constructive , totally disconnected space , inductive dimension , minkowski–bouligand dimension , hausdorff dimension , set (abstract data type) , discrete mathematics , path (computing) , euclidean distance , hausdorff space , unit interval , fractal dimension , locally compact space , fractal , computer science , mathematical analysis , geometry , process (computing) , programming language , operating system

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

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