---
tag: D9RY4667K6
kind: concept
name: Directed set
msc: [06A06, 54A20]
specializes: [0VABYWJWH9]
sources:
  - url: https://en.wikipedia.org/wiki/Directed_set
same_as:
  wikidata: Q1513048
  wikipedia: Directed set
---

## Summary

A directed set is a nonempty [[preordered set|0VABYWJWH9]] in which every two elements have a common upper bound.
Directed sets are the index sets of [[nets|4DVJQJV0DG]], which generalize sequences in topology, and of direct limits in algebra.
They also underlie directed-complete partial orders in domain theory.

## Explanations

### L1 human

In a directed set, any two elements can be "caught up with" by a third that is above both.
So the set has a sense of going further and further in one direction, even if it is not a line.
The natural numbers are directed, and so are the finite subsets of a set ordered by inclusion: $A \cup B$ is above both $A$ and $B$.
A set with two incomparable elements and nothing else is not directed.