---
tag: 75EJ0HB40M
kind: definition
name: Cofinal subset
msc: [06A06, "03E04"]
defines: 2PFQY2RZBC
uses: [0VABYWJWH9, 0BJR9QR08E]
sources:
  - url: https://en.wikipedia.org/wiki/Cofinal_(mathematics)
---

## Definition

Let $(P, \le)$ be a [[preordered set|0VABYWJWH9]] and let $C \subseteq P$ be a [[subset|0BJR9QR08E]].
The subset $C$ is *cofinal* in $P$ when for every $x \in P$ there is $c \in C$ with $x \le c$.

## Remarks

Example: the set $\N$ is cofinal in $\R$ with the usual order.
Check: for $x \in \R$, the integer $\ceil{\abs{x}}$ is in $\N$ and is at least $x$.

Non-example: the set $(-\infty, 0]$ is not cofinal in $\R$, since no element of it is at least $1$.

If $P$ has a greatest element $m$, then a subset is cofinal exactly when it contains some $c$ with $m \le c$.
In a partial order, this means that the subset contains $m$.
Being cofinal is transitive: a cofinal subset of a cofinal subset of $P$ is cofinal in $P$.
The dual notion is a coinitial subset, in which every element of $P$ lies above some element of the subset.