Definition:Cofinal Subset

Definition
Let $\left({S, \mathcal R}\right)$ be a relational structure, that is, a set $S$ endowed with a binary relation $\mathcal R$.

Let $T \subseteq S$ be a subset of $S$.

Then $T$ is a cofinal subset of $S$ with respect to $\mathcal R$ iff:
 * $\forall x \in S: \exists t \in T: x \mathop {\mathcal R} t$

Also known as
If the binary relation $\mathcal R$ is understood, then it is commonplace to omit reference to it.

A cofinal subset of $S$ (with respect to a given relation) can also be referred to as cofinal in $S$.

Also defined as
Although the definition pertains to arbitrary binary relations over $S$, in practice the notion of a cofinal set goes along with a partial ordering or a preorder.

Also see

 * Subset of Natural Numbers is Cofinal iff Infinite


 * Strictly Increasing Infinite Sequence of Positive Integers is Cofinal in Natural Numbers