Metric dimension

A numerical characteristic of a compact set, defined in terms of coverings of "standard measure" , the number of which defines the metric dimension. Let be a compact set, and let be the minimal number of sets with diameter not exceeding that are needed in order to cover . This function, depending on the metric in , takes integer values for all , and increases without bound as ; it is called the volume function of . The metric order of the compact set is the number

This quantity is not yet a topological invariant. Thus, the metric order of a curve in the sense of Jordan (cf. Line (curve)) with the Euclidean metric is equal to 1, but for a curve in the sense of Jordan passing through a perfect totally-disconnected set in of positive measure, this value is equal to . However, the greatest lower bound of the metric orders for all metrics on (called the metric dimension) is equal to the Lebesgue dimension (the Pontryagin–Shnirel'man theorem, 1931, see [1]).

References

[1]	W. Hurevicz, G. Wallman, "Dimension theory" , Princeton Univ. Press (1948)

Comments

Metric dimension makes sense for non-compact separable metrizable spaces (using totally bounded metrics), and the Pontryagin–Shnirel'man theorem extends to them. This was shown by E. Szpilrajn-Marczewski. See [a2].

There are also other types of metric-dependent dimension functions.

One example is the Hausdorff dimension.

Another example is obtained by modifying the definition of the covering dimension (see Dimension): If is a metric space, one defines by if and only if for every there is an open covering of with and . Here and means that no point of is an element of more than elements of . One can show that and that these inequalities are best possible, see [a1].

References

[a1]	R. Engelking, "Dimension theory" , North-Holland & PWN (1978) pp. 19; 50
[a2]	J.-I. Nagata, "Modern dimension theory" , Interscience (1965)