Subcountability

Summary

In constructive mathematics, a collection $X$ is subcountable if there exists a partial surjection from the natural numbers onto it. This may be expressed as

$$\exists (I\subseteq{\mathbb N}).\, \exists f.\, (f\colon I\twoheadrightarrow X),$$

where $f\colon I\twoheadrightarrow X$ denotes that $f$ is a surjective function from $I$ onto $X$. The surjection is a member of ${\mathbb N}\rightharpoonup X$ and here the subclass $I$ of ${\mathbb N}$ is required to be a set. In other words, all elements of a subcountable collection $X$ are functionally in the image of an indexing set of counting numbers $I\subseteq{\mathbb N}$ and thus the set $X$ can be understood as being dominated by the countable set ${\mathbb N}$.