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

Any strictly increasing sequence of integers $\Gamma:=\left\{ n_k\right\}_{k\in\N}\subseteq \N$ is a cofinal set of $\N$.

Thus, for a given sequence $\left\{x_n\right\}_{n\in\N}\subseteq \mathcal{X}$ (where $\mathcal{X}$ is any set), the set $\left\{x_\gamma\right\}_{\gamma\in\Gamma}$ is a subsequence of $\left\{x_n\right\}_{n\in\N}$.

It is straightforward to identify the counterpart of subsequences for nets: Let $\langle x_\lambda \rangle_{\lambda\in\Lambda}\subseteq \mathcal{X}$ be a net in $\mathcal{X}$ where $\Lambda$ is a partially ordered set. The counterparts of a "subsequence" of $\langle x_\lambda \rangle_{\lambda\in\Lambda}$ are sets of the form $\langle x_\sigma \rangle_{\sigma\in\Sigma}$ where $\Sigma$ is a cofinal subset of $\Lambda$.