metric entropy

Given a metric space $(X,\rho )$, the metric entropy is $\log N(\delta ;X,\rho )$ where $N(\delta ;X,\rho )$ is the $\delta$-covering number of $(X,\rho )$; see covering and packing.

Metric entropy is a measure of the complexity of the set $X$. Typically $N(\delta )$ diverges as $\delta \to 0$, and we are interested in the rate at which it goes to zero.