# Definition:Cofinal Subset

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## 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 $\mathcal R$** if and only if:

- $\forall x \in S: \exists t \in T: x \mathop {\mathcal R} t$

## 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.