---
tag: 8F933HJ1RY
kind: definition
name: Directed set
msc: [06A06, 54A20]
defines: D9RY4667K6
uses: [0VABYWJWH9]
sources:
  - url: https://en.wikipedia.org/wiki/Directed_set
---

## Definition

Let $D$ be a set with a [[preorder|0VABYWJWH9]] $\le$.
The set $D$ is *directed* (upward directed) when:

1. $D$ is nonempty.
2. For all $a, b \in D$ there is $c \in D$ with $a \le c$ and $b \le c$, that is, every two elements have an upper bound in $D$.

## Remarks

Example: the set of neighbourhoods of a point $x$ in a topological space, ordered by reverse inclusion ($U \le V$ when $V \subseteq U$), is directed.
Check: $x$ has the neighbourhood consisting of the whole space, and for neighbourhoods $U, V$ the set $U \cap V$ is a neighbourhood with $U \le U \cap V$ and $V \le U \cap V$.

Non-example: the set $\set{a, b}$ with the trivial order, where each element is only below itself, fails clause 2, since $a$ and $b$ have no common upper bound.

By induction, every finite subset of a directed set has an upper bound in it.
A downward directed set is defined dually, with lower bounds.
Some authors, and the Mathlib predicate `IsDirected`, drop clause 1.