---
tag: YW8YSCFPG5
kind: concept
name: Club set
msc: ["03E05", "03E10"]
sources:
  - url: https://en.wikipedia.org/wiki/Club_set
same_as:
  wikidata: Q310647
  wikipedia: Club set
---

## Summary

A club set is a subset of a limit ordinal that is closed under limits and unbounded, and the name abbreviates "closed unbounded".
Below a [[regular cardinal|13ZV2F8JAR]] $\kappa > \omega$, the club sets play the role of "large" sets: the intersection of fewer than $\kappa$ of them is again club.
They define the [[stationary sets|YE6WZR46RQ]], and the [[precise definition|GWS5QXH7WW]] is given separately.

## Explanations

### L1 human

Think of the countable ordinals as a very long line.
A club set is a subset that keeps appearing as far as you go, and that contains every point it approaches from below.
For example, the limit ordinals below $\omega_1$, such as $\omega$, $\omega \cdot 2$ and $\omega^2$, form a club set.
The successor ordinals do not: they keep appearing, but the limits they approach, like $\omega$, are not successors.
Two club sets of countable ordinals always meet, and in fact their intersection is again club.