Skeletons and Central Sets |
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Authors: | Fremlin DH |
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Institution: | Department of Mathematics, University of Essex Wivenhoe Park, Colchester CO4 3SQ, Essex, UK E-mail: fremdh{at}essex.ac.uk |
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Abstract: | Let be an open proper subset of Rn. Its skeleton is the setof points with more than one nearest neighbour in the complementof its central set is the set of centres in maximal open ballsincluded in . Intuitively, if we think of as a land mass inwhich height is proportional to distance from the sea, its skeletonand central set can be thought of as corresponding to ridgesin the mountains of . In this note I discuss the metric andtopological properties of such sets. I show that any skeletonin Rn is F , and has dimension at most n 1, by any ofthe usual measures of dimension; that if is bounded and connected,its skeleton and central set are connected; and that separatesRn iff its skeleton does iff its central set does. Any centralset in Rn is a G set of topological dimension at most n 1. In the plane, I show that both skeletons and central setsare locally path-connected, and indeed include many paths offinite length. For any , its central set includes its skeleton;I give examples to show that the central set can be significantlylarger than the skeleton. 1991 Mathematics Subject Classification:54F99. |
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Keywords: | skeleton central set path-connected rectifiable-path-connected |
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