---
tag: ZR8A03FV67
kind: concept
name: Scott topology
msc: [06B35, 06B30, 54A10]
sources:
  - url: https://en.wikipedia.org/wiki/Scott_continuity
---

## Summary

The Scott topology on a partially ordered set has as open sets the up-closed sets that cannot be reached by a directed supremum from outside.
It turns a [[directed-complete partial order|7TSXP95QN6]] into a topological space whose continuous maps are the Scott-continuous maps.
Its [[specialization preorder|CMYVK9CEKZ]] is the original order, so the order can be recovered from the topology.
It is the topology used in [[domain theory|Q2P5D74Q1V]].

## Explanations

### L1 human

An open set in the Scott topology is an "observable property" of partial information.
If an element has the property, then every better-informed element has it too, so open sets are closed upward.
And if a limit of approximations has the property, then some approximation already has it, since one can only observe finitely much.
For example, in $\N \cup \set{\infty}$, the set $\set{\infty}$ is not Scott-open, because $\infty$ is the supremum of $0, 1, 2, \ldots$, none of which is in the set; but every set $\set{n, n + 1, \ldots, \infty}$ is Scott-open.