---
tag: G2QNFT210W
kind: concept
name: Locale
msc: [06D22, 18F70, 54A05]
sources:
  - url: https://en.wikipedia.org/wiki/Pointless_topology
---

## Summary

A locale is a generalized topological space described only by its lattice of "open sets", which is a [[frame|306PWGH7BF]].
Maps of locales go in the opposite direction to frame homomorphisms, just as a continuous map pulls open sets back.
Every topological space gives a locale, and for [[sober spaces|4N4DNDR22Z]] the space can be recovered from it.
Locales are the objects of [[pointless topology|CYEMGH98WN]].

## Explanations

### L1 human

A continuous map $f \colon X \to Y$ sends points forward but sends open sets backward, by taking preimages.
If one only remembers the open sets, it is natural to describe a map of spaces by a map of open-set lattices going the other way.
A locale is a frame regarded in this reversed way, as a space.
The real line, for instance, gives the locale whose frame is the set of open subsets of $\R$.
There are also nontrivial locales with no points at all, which have no counterpart among ordinary spaces.