Definition:Cofinal Subset

Definition

Let $\struct {S, \RR}$ be a relational structure, that is, a set $S$ endowed with a binary relation $\RR$.

Let $T \subseteq S$ be a subset of $S$.

Then $T$ is a cofinal subset of $S$ with respect to $\RR$ if and only if:

$\forall x \in S: \exists t \in T: x \mathrel \RR t$

Also known as

If the binary relation $\RR$ 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

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): cofinal