---
tag: ZC8DNB75N4
kind: concept
name: Lawson topology
msc: [06B35, 54F05]
specializes: [0CQHFPGJ45]
sources:
  - url: https://en.wikipedia.org/wiki/Lawson_topology
same_as:
  wikidata: Q6505020
  wikipedia: Lawson topology
---

## Summary

The Lawson topology on a partially ordered set is the coarsest topology that contains both the [[Scott topology|ZR8A03FV67]] and the lower topology.
The Scott topology sees only the "upward" behaviour of the order, so it is rarely Hausdorff.
Adding the lower topology repairs this: on a [[continuous lattice|RXBD58SGXZ]] the Lawson topology is compact and Hausdorff.
It is the main tool for treating domains and continuous lattices as genuine compact spaces.

## Explanations

### L1 human

The Scott topology has only open sets that are closed upward, so a point cannot be separated from the points above it.
The lower topology has the opposite bias: its basic open sets are the points that are not above some fixed elements.
The Lawson topology lets both kinds of sets be open, so it can separate points in both directions.
On the unit interval $[0, 1]$ with its usual order, the proper nonempty Scott open sets are the intervals $(a, 1]$.
The lower topology is generated by the intervals $[0, b)$.
Together they give the usual topology of $[0, 1]$.