---
tag: 2PFQY2RZBC
kind: concept
name: Cofinal subset
msc: [06A06, "03E04"]
sources:
  - url: https://en.wikipedia.org/wiki/Cofinal_(mathematics)
same_as:
  wikidata: Q5141045
  wikipedia: Cofinal (mathematics)
---

## Summary

A subset $C$ of a preordered set $P$ is cofinal when every element of $P$ lies below some element of $C$.
A cofinal subset reaches arbitrarily far up in $P$.
For example, $C$ and $P$ have the same upper bounds, so a supremum of $P$ can be computed from $C$ alone.
The least cardinality of a cofinal subset is the [[cofinality|BAWHE1W6MH]] of $P$.

## Explanations

### L1 human

Think of $P$ as heights you can reach, and $C$ as a set of ladders.
The set $C$ is cofinal when, for any height, some ladder goes at least that high.
In the natural numbers, the even numbers are cofinal, and so are the powers of $2$.
A finite set of natural numbers is never cofinal in $\N$, because nothing in it is above its largest element plus one.