---
tag: CYEMGH98WN
kind: concept
name: Pointless topology
msc: [06D22, 18F70, 54A05]
sources:
  - url: https://en.wikipedia.org/wiki/Pointless_topology
same_as:
  wikidata: Q7208386
  wikipedia: Pointless topology
---

## Summary

Pointless topology studies spaces through their lattices of open sets instead of through their points.
Its basic objects are frames, complete lattices in which finite meets distribute over arbitrary joins, and locales, which are frames viewed as generalized spaces.
Every topological space gives a locale, its frame of open sets, and for [[sober spaces|4N4DNDR22Z]] no information is lost.
Some theorems that need the axiom of choice for spaces, such as the product theorem for compact spaces, hold for locales without it.

## Explanations

### L1 human

In ordinary topology a space is a set of points together with a choice of open sets.
Pointless topology forgets the points and keeps only the open sets, with their order by inclusion, their unions and their finite intersections.
The open sets of the real line, for example, form such a structure, and the points can be recovered from it.
Working without points makes some constructions simpler and more constructive.
It also allows "spaces" that have open sets but no points at all.