---
tag: 9EE43J2W16
kind: definition
name: Way-below relation
msc: [06B35, 06A06]
defines: RDM7J5J9HQ
uses: [0QV2JS9RJ3, D9RY4667K6, 0R3H5PANRQ, 9F7TYES4KE]
sources:
  - url: https://en.wikipedia.org/wiki/Domain_theory
---

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]] and let $x, y \in P$.
The element $x$ is *way below* $y$, written $x \ll y$, if the following holds.
For every [[directed subset|D9RY4667K6]] $D \subseteq P$ with a [[supremum|0R3H5PANRQ]] such that $y \le \bigvee D$, there is $d \in D$ with $x \le d$.

## Remarks

If $x \ll y$, then $x \le y$, by taking $D = \set{y}$.
If $w \le x \ll y \le z$, then $w \ll z$.
An element $c$ is [[compact|9F7TYES4KE]] if and only if $c \ll c$.
Example: in $[0, 1]$ with the usual order, $x \ll y$ holds if and only if $x = 0$ or $x < y$; for $x < y \le \bigvee D$, some $d \in D$ exceeds $x$ by the definition of the supremum.
Non-example: in $[0, 1]$, $1/2 \not\ll 1/2$, because $D = \set{1/2 - 1/n : n \ge 3}$ is directed with supremum $1/2$ and no member is at least $1/2$.