Note on Metric Dimension

The metric dimension of a compact metric space is defined here as the order of growth of the exponential metric entropy of the space. The metric dimension depends on the metric, but is always bounded below by the topological dimension. Moreover, there is always an equivalent metric in which the metric and topological dimensions agree. This result may be used to define the topological dimension in terms of the metric entropy as the infimum of the metric dimension taken over all metrics. Kolmogorov's concept of e-entropy, defined for compact metric spaces [3], has proved a useful measure of the "bulk" of such a space. In this note we relate this concept to the Hausdorff and topological dimensions of spaces for which these dimensions are defined. Our result appears as Theorem 2 below. Let X be a compact metric space, with distance function p. If U is any subset of X, the diameter of U is the finite number (1) 8(U) = sur>{p(x,y):x,yEU}. If 'U. is any family of subsets U of X, then the mesh of It is the number (2) ix = sup{5(c7):t/£0 there exists a finite open cover of mesh0 there exists a finite open cover {Ui\ of X such that yjl, b~(Ui)v<t). In this case we put (4) hd(Z) = ini{p:h-dim(X) g p] Received by the editors May 23, 1969. AMS Subject Classifications. Primary 5435, 5470; Secondary 4141.