Vol. 23, No. 1, 1967

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ISSN: 0030-8730
On the relationship between Hausdorff dimension and metric dimension

Albert Chapman Vosburg

Vol. 23 (1967), No. 1, 183–187
Abstract

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
Milestones
Received: 24 October 1966
Published: 1 October 1967
Authors
Albert Chapman Vosburg