Definition:Cofinal Relation on Ordinals

Definition

Let $x$ and $y$ be ordinals.

Then $y$ is said to be cofinal with respect to $x$ if and only if there exists a mapping $f: y \to x$ such that:

\((1)\)	$:$							$y \le x$
\((2)\)	$:$							$f$ is strictly increasing.
\((3)\)	$:$							For all $a \in x$, there is some $b \in y$ such that $\map f b \ge a$

Notation

If $y$ is cofinal with $x$, the notation $\map {\mathrm {cof} } {x, y}$ can be used.

Warning

$\mathrm {cof}$ is not symmetric. In fact, it is antisymmetric.

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Also see

• Definition:Cofinal Subset

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 10.51$