Definition:Cofinal Subset

Definition
Let $\left({\mathcal X, \preceq}\right)$ be a relational structure, that is, a set $\mathcal X$ endowed with a binary relation $\preceq$ (usually a partial ordering).

A subset $\Sigma \subseteq \mathcal X$ is said to be a cofinal subset of $\mathcal X$ iff:
 * $\forall x \in\mathcal X: \exists \sigma \in \Sigma: x \preceq \sigma$

That is for every $x$ in $\mathcal X$ there is a $\sigma$ in $\Sigma$ such that $x$ relates to $\sigma$.

Note
Although the definition pertains to arbitrary binary relations over $\mathcal X$, in practice the notion of a cofinal set goes along with a partial order or a preorder.

The notion of cofinal sets has a special place in the study of nets and filters.

Also see

 * Cofinal Sets of Natural Numbers