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

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

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

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