  We introduce the so--called doubling metric on the collection of non--empty
bounded open subsets of a metric space. Given a subset $U$ of a metric space
$X$, the predecessor $U_{*}$ of $U$ is defined by doubling the radii of all
open balls contained inside $U$, and taking their union. If $U$ is open, the
predecessor of $U$ is an open set containing $U$. The directed doubling
distance between $U$ and another subset $V$ is the number of times that the
predecessor operation needs to be applied to $U$ to obtain a set that contains
$V$. Finally, the doubling distance between $U$ and $V$ is the maximum of the
directed distance between $U$ and $V$ and the directed distance between $V$ and
$U$.

General Topology;Classical Analysis and ODEs;Metric Geometry