# 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 $F$ be a compact set, and let $N _ {F} ( \epsilon )$ be the minimal number of sets with diameter not exceeding $\epsilon$ that are needed in order to cover $F$. This function, depending on the metric in $F$, takes integer values for all $\epsilon > 0$, and increases without bound as $\epsilon \rightarrow 0$; it is called the volume function of $F$. The metric order of the compact set $F$ is the number
$$k = fnnme \underline{lim} \ \left ( - \frac{ \mathop{\rm ln} N _ {F} ( \epsilon ) }{ \mathop{\rm ln} \epsilon } \right ) .$$
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 $\mathbf R ^ {n}$ of positive measure, this value is equal to $n$. However, the greatest lower bound of the metric orders for all metrics on $F$( called the metric dimension) is equal to the Lebesgue dimension (the Pontryagin–Shnirel'man theorem, 1931, see ).