---
tag: QW999ZFZVN
kind: definition
name: Lawson topology
msc: [06B35, 54F05]
defines: ZC8DNB75N4
uses: [0QV2JS9RJ3, ZR8A03FV67, 0CQHFPGJ45, 09S8DV34Y0]
sources:
  - url: https://en.wikipedia.org/wiki/Lawson_topology
---

## Definition

Let $(P, \le)$ be a [[partially ordered set|0QV2JS9RJ3]].
For $x \in P$, write ${\uparrow} x = \set{y \in P : x \le y}$.
The *lower topology* on $P$ is the topology with [[subbasis|09S8DV34Y0]] the sets $P \setminus {\uparrow} x$ for $x \in P$.
The *Lawson topology* on $P$ is the smallest [[topology|0CQHFPGJ45]] on $P$ that contains both the [[Scott topology|ZR8A03FV67]] and the lower topology.
Equivalently, it is the topology generated by the union of these two topologies as a subbasis.

## Remarks

Example: on $P = [0, 1]$ with the usual order, the Scott open sets are $\emptyset$, $[0, 1]$ and the intervals $(a, 1]$ with $0 \le a < 1$.
The lower open sets are generated by the intervals $P \setminus {\uparrow} b = [0, b)$.
Finite intersections of these sets give all intervals $(a, b)$, $[0, b)$ and $(a, 1]$, so the Lawson topology is the usual topology of $[0, 1]$.

Non-example: the Scott topology of $[0, 1]$ is not its Lawson topology.
The set $[0, 1/2)$ is Lawson open, but it is not an upper set, so it is not Scott open.

The Lawson topology of a [[continuous lattice|RXBD58SGXZ]] is compact and Hausdorff.
Some authors define the Lawson topology only for directed-complete partial orders or for continuous posets, where it is best behaved.