Definition:Cofinal Subset

Definition
Let $(\mathcal{X},\leq)$ be a set endowed with a binary relation (e.g. a partial order).

A set $\Sigma\subseteq \mathcal{X}$ is said to be a cofinal subset of $\mathcal{X}$ if for every $x\in\mathcal{X}$ there is a $\sigma\in\Sigma$ such that $x\leq \sigma$.

Cofinal Sets of $\N$
For the partially ordered space $(\N,\leq)$, a set $\Sigma\subseteq \N$ is cofinal if and only if it is infinite.