---
tag: ZV38HA5PAQ
kind: concept
name: Stone space
msc: ["06E15", "54D30"]
specializes: [0TVJZYB6SH]
sources:
  - url: https://en.wikipedia.org/wiki/Stone_space
same_as:
  wikidata: Q30587585
  wikipedia: Stone space
---

## Summary

A Stone space is a [[compact|0TVJZYB6SH]] [[Hausdorff|0S1FJ8VEX9]] [[totally disconnected|0RJSBR9GW1]] topological space.
Stone spaces are also called Boolean spaces or profinite spaces.
They are exactly the spaces of ultrafilters of Boolean algebras, by [[Stone's representation theorem|S4QJ395PA5]], and [[Stone duality|B9T33D5MJK]] makes the two theories equivalent.
The Cantor set and the profinite groups are standard examples.

## Explanations

### L1 human

A Stone space is compact, so it is "small" in the topological sense, but it is also completely broken into dust: no two points are connected.
Its open sets are generated by sets that are open and closed at the same time.
The Cantor set is the typical picture: it is closed and bounded, yet any two of its points can be split by a gap.
Every finite set with the discrete topology is a Stone space too.