---
tag: 9F7TYES4KE
kind: concept
name: Compact element
msc: [06B23, 06B35]
sources:
  - url: https://en.wikipedia.org/wiki/Compact_element
same_as:
  wikidata: Q5155306
  wikipedia: Compact element
---

## Summary

A compact element of a partially ordered set is an element with this property.
Whenever it lies below the supremum of a directed set, it already lies below some member of that set.
In a complete lattice this means: whenever it lies below a join, it lies below the join of finitely many of the joined elements.
Compact elements play the role of finite or finitely generated objects, and they are the building blocks of [[algebraic lattices|3J0BTXSCVN]] and of [[Scott domains|1SAAZG46DG]].

## Explanations

### L1 human

The name comes from compact sets in topology, where every open cover has a finite subcover.
In the lattice of subsets of a set, an element is compact exactly when it is a finite set.
A finite set inside a union of sets is already inside the union of finitely many of them.
In the lattice of subgroups of a group, the compact elements are the finitely generated subgroups.
So compactness is the order-theoretic way to say "finite" without counting.