---
tag: 833XAA2Q65
kind: concept
name: Regular open set
msc: [54A05]
specializes: [02BH3MCBQB]
sources:
  - url: https://en.wikipedia.org/wiki/Regular_open_set
same_as:
  wikidata: Q110745661
  wikipedia: Regular open set
---

## Summary

A regular open set in a [[topological space|0CQHFPGJ45]] is an [[open set|02BH3MCBQB]] that equals the [[interior|0RZH6W98HK]] of its [[closure|0QEAJB91GZ]].
Such a set has no "cracks": it does not miss single points or thin pieces inside its own closure.
The regular open sets of any space form a complete [[Boolean algebra|0XADJKM699]], which makes them useful in set theory and in pointless topology.

## Explanations

### L1 human

Take an open set, fill in its boundary by taking the closure, and then remove the boundary again by taking the interior.
If you get back the same set, the set is regular open.
An open interval $(0, 1)$ in $\R$ is regular open: closing gives $[0, 1]$, and the interior of that is $(0, 1)$ again.
The set $(0, 1) \cup (1, 2)$ is not, because the process fills the gap at $1$ and returns $(0, 2)$.