# Covering number

In mathematics, the idea of a *covering number* is to count how many small spherical balls would be needed to completely cover (with overlap) a given space. There are two closely related concepts as well, the *packing number*, which counts how many disjoint balls fit in a space, and the *metric entropy*, which counts how many points fit in a space when constrained to lie at some fixed minimum distance apart.

## Mathematical Definition

More precisely, consider a subset of a metric space and a parameter . Denote the ball of radius centered at the point by . There are two notions of covering number, internal and external, along with the packing number and the metric entropy.

- The
**packing number**is the largest number of points such that the balls fit within K and are pairwise disjoint. - The
**internal covering number**is the fewest number of points such that the balls cover . - The
**external covering number**is the fewest number of points such that the balls cover . - The
**metric entropy**is the largest number of points such that the points are -separated, i.e. for all .

## Inequalities and Monotonicity

The internal and external covering numbers, the packing number, and the metric entropy are all closely related. The following chain of inequalities holds for any .^{[1]}

In addition, the quantities are non-increasing in and non-decreasing in for each of . However, monotonicity in can in general fail for .