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On the relationship
between Hausdorff dimension and metric dimension
Albert Chapman Vosburg |
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Vol. 23 (1967), No. 1, 183–187
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Abstract |
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The definitions of the Hausdorff dimension dimhX, upper metric dimension dimX and lower metric dimension dimX of a metric space X all depend upon asymptotic characteristics of diameters of sets in covers of X. We relate these notions. First we note that dimhX ≤ dimX holds for all totally bounded metric spaces X, while on the other hand there exist perfect subsets A of [0,1] satisfying dimhA = 0 and dimA = 1 = dim[0,1]. Finally we show that there exist perfect subsets S of [0,1] which satisfy dimhS = 0 and dimS = 1 even when strong local conditions are imposed. |
Mathematical Subject Classification
Primary: 54.70
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Milestones
Received: 24 October 1966
Published: 1 October 1967
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Authors |
| Albert Chapman Vosburg | |
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