Countability of pairwise disjoint open disks ($|\mathcal{D}|\le {\aleph }_0$)

Every family of pairwise disjoint nonempty open disks in the plane is countable. The density of the rational numbers gives each disk a point of the countable set ${\mathbb{Q}}^2$. Fixing an enumeration and taking the first point in each disk yields an injection from disks to labels. This argument requires no countable choice of representatives and is useful when geometric separation would otherwise produce an uncountable family.